Fist off, just to get it out the way, I think Ynneadwraith was right when they say the Blanks are more than just soulless. Things like rocks and water are soulless, but Blanks are the utter anathema to the warp. It's not that Blanks dont exist in the Warp, the Warp cannot exist around Blanks. If that was simply because a Blank is soulless then rocks, water, and the rest of it would also be anathema to the Warp, making Daemonic invasions and the use of psychic powers damn near impossible. So it's more accurate to say that Blank's have an anti-soul of sorts; a soul made from something other than psychic energy. The reason psykers describe them as soulless is because they cannot see this other soul.

This leads me into my thoughts on the Necrontyr soul. I have always seen the soul to be some sort of energy, as many authors often refer to it as the "life energy" of a creature. Indeed, when the 5th ed. Necron Codex refers to the biotransference of the Necrontyr it describes the C'tan as "drinking in the torrent of cast-off life energy" and that Szarekh "watched the C'tan feast of the life essence of his people" before later realising that "the price of physical immortality had been the loss of his soul" (page 7 of the 5th edition Necron codex). More to this, we know the C'tan are star vampires and that they fed off the energy of the stars before discovering the delicacy that is a soul, so this is further evidence (just from Necron sources) that a soul is energy. But does it have to be psychic energy, linked to the warp?

Most races in the 41st millennium have a soul of psychic energy, but as I said Blanks appear to have a different, anti-psychic kind of soul, so soul-energy must not be limited to psychic energy. However the most important thing about the Necrontyr is they are NOT a race of the 41st millennium. They are truly ancient: before the War In Heaven they already had incredibly advanced technology and science (they had living metal before they met the C'tan) and had carved out a significant empire in the galaxy; then the War In Heaven started and lasted centuries, if not millennia (5th ed codex (page 6) says that the Necrontyr's pre-C'tan war against the Old Ones saw them pushed back in "but a span of centuries" but says nothing about the second half, so I'm assuming a bit here); and then after the war the Necrons slept for 60 million years.

My point is that the Necrontyr are OLD, and the galaxy was a very different place back then. Despite the fluff surrounding this time becoming scarcer and scarcer, one thing seems to remain consistent: before the War in Heaven, psychic power wasn't really a thing beyond the Old Ones. Certainly, we know that the Necrontyr either lacked psychic abilities entirely or the gene was so withered within their DNA that it was all but useless, hence why the could not defeat the Old Ones without the help of the C'tan. Even though they had superior numbers and technology, they just could not counter psykers effectively.

So we have a galaxy in which the only major psychic presence is the Old Ones. But then the War In Heaven begins and eventually the Necrons and the C'tan start winning. In desperation the Old Ones weaponize not just psychic powers, but the psyker gene. They begin manipulating and elevating other races they have been nurturing to fight alongside them. Specifically, they nurture the psyker gene in these races, realising that it is their greatest weapon against the psyker-less Necrons and C'tan. Among the many races, they raise the Eldar to become a psychically mighty fighting force, but they cannot replace their numbers fast enough to match the Necrons. To answer this they enhance the Krorks/Orks to be a rapidly multiplying combat-ready force, born to instinctively know how to make/fix/fight things and with a gestalt psychic field to ensure their technology works. They even begin work on evolving humans according to some sources, but lose the war before they can finish which is why it took so long for the psyker gene to fully develop in us. Now the galaxy is saturated with the psyker gene, and psychic races with psychic souls. These races make up the majority of what we see in the 41st millennium, but the Necrontyr/Necrons are not one of them. The Necrontyr came before these races and weren't nurtured by the Old Ones. They didn't have the psyker gene thrust upon them because they were the threat it was answering.

And this long meandering explanation bring me to my point. Without the psychic gene, there's not logical reason that the Necrontyr would have souls of psychic energy that are linked to the warp. They certainly had some sort of soul-energy since the C'tan definitely consumed it, but it likely took another form. What form this energy did take is anyone's guess. It could have been limited to the Materium, or it could have worked similar to psychic energy and been linked to another dimension (we know from Flayed Ones, Deathmarks, and Nebuloscopes that their are more dimensions than just realspace and the Warp). If it was linked to another dimension that could explain why the warp never really interact with it (we don't know how the warp interacts with any dimension beyond realspace), and if the warp can't directly interact with the soul, then it can't easily posses the mind and body of the creature that soul belongs/once belonged to.

To me this is less about an individual Necron wanting to return to a body of flesh and blood, and more about the Necrons as a species wanting to return to bodies of flesh and blood.

The Necrons have no souls (I know the whole point of this thread is to argue that point but lets just assume it is either gone or unrecognisable mutilated). They were devoured by the C'tan and now those with enough sentience feel indescribably hollow, there is a void of nothingness that can never be filled or satisfied. Though the Overlords and Pharons are too proud and arrogant to admit it, they feel that they suffer from a terrible curse. With their souls gone the people and culture of the Necrontyr are all but destroyed, replaced by the soulless warmachine that is Necrons. If the Necrons can return to flesh bodies, then maybe there is a chance their species will regain their souls (if not in those who were once Necrons then in their children). They can cast of the memories of the C'tan and their servitude to them, which is arguably a time of great shame for such a vain race. Like Psienesis said, the Necrontyr sacrificed all but the memory of their past civilisation when they underwent biotransference. With flesh bodies they can grow their species once more. They can revive their culture and return it to it's past glories. More importantly, they can fill that hollow emptiness.

Beyond this, there is the important factor which is that the Necrons cannot reproduce. Yes they are incredibly durable, but once a Necron is destroyed beyond the ability to be repaired, that is the Necron population down by 1 permanently. If they have flesh bodies, they can reproduce again, bringing new life to their species and allowing it to grow and advance once more, not stagnate as it does now.

Warriors might not have enough sentience to survive outside of their Living Metal bodies anymore, but the rest of the Necrons probably could. Immortals would likely suffer from severe cognitive limitations but the Pharons, Overlords, Lords, Crypteks, Lychguard and Praetorians still make up a sizeable population. Keeping the warriors in Living Metal bodies, they could definitely remain a threat to the other races of the galaxies, which brings me to my next point.

Beyond the above mentioned wanting to get flesh bodies again, this really comes down to two simple yet defining aspects of the Necron psyche: pride and arrogance.

They are the Necrons. One of the first races to rule the galaxy. They carved out a vast empire before many of todays races even had a homeworld to evolve on. They went to war with two of the most powerful races in the galaxy, which could each bend reality to their will, and not only won but utterly destroyed them. They killed their own gods and took their mantle. The Necrons wont let the Tyranids "roflstomp the galaxy for them" because it's not the galaxy, it's their galaxy. As far as the Necrons are concerned the galaxy and everything in it is their property, and the Tyranids not only threaten that, but insult the Necrons in daring to do so.

Also, what good is lording over the galaxy if there is no one in it to lord over? Earlier this year I read The World Engine by Ben Counter, and whilst I'll admit that there is a lot of fluff in it (both concerning Necrons and Space Marines) that I did not like, it had a lot of points built upon the Newcron fluff which actually made me like it (especially the stuff The Judicator talks about). One of these points was that the Necrons had taken slaves from the worlds the World Engine had destroyed. Now I will admit that when I first read this I thought it was utterly ridiculous. Why would the the Necrons need slaves when they have the Canoptek units? Why would they allow life which is so far below them to serve them. Again the answer lies in pride and arrogance. What use is it having legions of warrios and hordes of Canoptek units serving you if you know that they have no concept of that fact that they are doing it? Canoptek units were made to serve, and the majority of Necrons which aren't high-ranking nobility have built in protocols which force them to serve. But if you enslave living, sentient races which know and understand you have overpowered them, then it can stroke your ego and satisfy your pride.

So beyond the Necrons wanting to return to the realm of flesh, they want to stop the Tyranids because this is their galaxy to conquer and subjugate.

In The World Engine, Ben Counter also built up another bit of fluff which I really want to see discussed further in the codexes. Its a return to the old notion of the Necrons harvesting the living, but takes it to new places since they no longer use conquered populations to make new Necrons. In chapter 3 the Astral Knights discover a room filled with machines deconstructing the remains of humans down to their base parts. I'm not just talking about dissecting here:

The first thing he saw was a human corpse, naked and skinny, hanging from a hook that rattled along a track built into the ceiling. Thin arms of glistening articulated steel pared the body apart, separating the head and the limbs. The parts were deposited on an elaborate conveyor which took them towards a bank of processing machines at one end of the room, with dozens of insect-like robot appendages to pick up the parts and force them down jaws lined with blades and grinders.
Tiny scarabs scurried across the conveyor as oozing body parts moved along. Their mandibles peeled off skin and muscle, separating organ and bone into neat piles arranged as precisely as the work of an artist. By the time the body reached the body reached the processors it had been broken down into pieces so completely that it did not resemble a person anymore, but the components of a puzzle from which a human corpse could be assembled.
...
A larger machine loomed past the first. This one received the processed corpse matter, carried by hundreds of scarabs. This machine, like a squat hundred-armed beast of steel, transformed the body parts into blocks of compressed bone or flesh, to be carried off by yet more scarabs to be stacked like so much cargo against the wall of the room.
Everything glistened with blood.

We are never explicitly told why the Necrons are doing this, but to me it obviously isn't to study out biology in an attempt to reverse the biotransference. They wouldn't so completely destroy the body of they were doing that. No, this is harvesting the living in a very literal sense, and perhaps much darker than what happened in the old fluff. They're not just killing and enslaving the other races of the galaxy, they are harvesting us like raw materials. In the same way that we dig up coal or cut down trees, the Necrons are disassembling organic life for production material. It goes without saying that Necron technology and science is beyond advanced compared to other races, so its not a stretch to assume that they can transform flesh and bone into fuel or living metal or some other useful thing.