The Knight's Romp

First things first: let's check the divisibility.

There are 64 squares, and the knight is standing in one of them. That leaves 63 squares to cover, and each move covers 3 squares, so that seems to work out. That means, however, that we won't be able to create a closed loop, so every solution we find only solves the puzzle for the starting point and, by reversing, the end point. (And the symmetrical counterparts of each all around the board.)

Then, let's see if we can find a path starting from some square. Choosing a starting point so that we can put one knight's move nicely in a corner (it feels like corners and nooks are going to be the most difficult to cover), it turns out that it's

This should convince us that there are no general proofs to the contrary, so now we just need to find an opposite case, or a handful of cases with the same result. (We are not going to need much more than that, because of the board's symmetry.)

That is, however, a lot trickier than it seems, every heuristic seems to have its exceptions, so I'm posting this as a partial answer, and will get back to it after lunch. (If anyone wants to continue from here, please do.)

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EDIT:

Here are two more of the similar kind:

So we are at least halfway there:

With Jaap's generous help, these were also found workable:

So we've managed to narrow the problem down quite a bit:

Usually, after this much fiddling, some pretty solid heuristics have emerged, but that's not the case here. If there's some clever logic that applies, it'll be on the order of "parities of parities don't match" or "every move is to another 2x2 square", but I'm not seeing any use for either of those, either. So my next approach would be a brute force search, which I'll have to leave for someone else to implement.