Book 7: When Lord Voldemort "killed" Harry in the Forbidden Forest, he is apparently knocked out (like Harry) for some reason.
What happened to Voldemort when he was knocked out? Did he also visit Limbo?
While he was unconscious, Voldemort was indeed in limbo, in the mutilated form that Harry and Dumbledore witness in King’s Cross Station. This is implied in the book, and was clearly stated in the answer to a question in the FAQ section of JKR’s old website.
What exactly was the mutilated baby-like creature Harry saw at King's Cross in chapter 35 of 'Hallows'?
I’ve been asked this a LOT. It is the last piece of soul Voldemort possesses. When Voldemort attacks Harry, they both fall temporarily unconscious, and both their souls - Harry's undamaged and healthy, Voldemort’s stunted and maimed - appear in the limbo where Harry meets Dumbledore.
- F.A.Q. on J.K. Rowling’s website
This is why Dumbledore tells Harry that he has less to fear from returning to King’s Cross - Harry will return whole, but Voldemort will return in that mutilated form.
“But I know this, Harry, that you have less to fear from returning here than he does.’
Harry glanced again at the raw-looking thing that trembled and choked in the shadow beneath the distant chair.”
That is also why Harry told Voldemort that he’s seen what he’ll become without remorse - in King’s Cross, Harry had seen what Voldemort’s soul would exist as in the afterlife unless Voldemort chose to repair it by having remorse for his actions.
“It’s your one last chance,’ said Harry, ‘it’s all you’ve got left … I’ve seen what you’ll be otherwise … be a man … try … try for some remorse …”
- Harry Potter and the Deathly Hallows, Chapter 36 (The Flaw in the Plan)
So yes, Voldemort did indeed enter limbo while unconscious.
However, what exactly caused Voldemort to fall unconscious and enter limbo to begin with is not clear. No reason for this is confirmed, so theories would be merely reasoned speculation. It is possible that it could have been because he used the Elder Wand against its true master. His Killing Curse destroyed the piece of his soul in Harry, but he did not usually fall unconscious when other pieces of his soul were killed in the destruction of his Horcruxes, so it being a result of that seems unlikely.
The others have covered what Voldemort experienced while unconscious, but not the why. It's not expressly laid out, but we can make some guesses. Harry was knocked out at the same time, too, and it is likely that it was for the same reason.
First, as Dumbledore explains in the vision of King's Cross Station, Voldemort's killing curse did not attack Harry's soul (or, to be more precise, his natural connection to his soul), but was channeled towards the piece of Voldemort's soul that Harry had been carrying for sixteen years. This is likely because while some of the benefits of Lily's sacrifice were no longer in effect (such as the protection gained from living in Petunia's house, or Voldemort being unable to touch Harry without experiencing pain), the basic protection from harm was still in effect.
Second, we know that Voldemort felt nothing when the soul pieces in the various Horcruxes were destroyed. This indicates that the soul piece in a properly-created Horcrux is neatly hemmed (so to speak) so that its only interaction with the wizard who created it is to tether the remaining piece of the wizard's soul to this world.
Third, we know that the piece of soul carried by Harry was not treated in this way, and so it still was able to affect Voldemort and be affected by him in various ways, such as Harry having access to Voldemort's thoughts.
This all adds up to the explanation that when Voldemort cast the Killing Curse against Harry, most of its force was expended against the lingering piece of soul that was in Harry, extinguishing it, but a small portion of the spell's force was divided between Harry and Voldemort, who were both connected to this fragment at the moment of its destruction, knocking both of them out.