Any games played of this here?
Variant idea: Veto Chess
I remember discussing this variant somewhere in 2008, during a computer competition. The easiest way to convert existing software into playing this is by changing the rules slightly in a non-essential way: instead of vetoing the move in advance you could veto it after it was made, by using the normal 'take back' function of the interface. The engine would then have to me modified such that after a takeback it won't reconsider that same move.

I remember discussing this variant somewhere in 2008, during a computer competition. The easiest way to convert existing software into playing this is by changing the rules slightly in a non-essential way: instead of vetoing the move in advance you could veto it after it was made, by using the normal 'take back' function of the interface. The engine would then have to me modified such that after a takeback it won't reconsider that same move.
Interesting idea, but the engine would still be using a non-optimal strategy. The move that is initially chosen and then vetoed is the move that is best under normal circumstances, which is not necessarily the same as the move that is optimal under the modified rules.
For example, a move that is a blunder under normal rules but only has one refutation might be a brilliant move in this variant, since the refutation could be vetoed. The engine would never consider such a move.
To make an engine play this with an optimal strategy for the modified rules, it would need to consider the veto possibility in its search algorithm.
Sure, the search of the engine would have to be modified too. But that would be the trivial part. Where engines normally calculate the score of a position as the score of the best move from it, it would have to take the score of the second-best move instead. As the best move would be vetoed.
It would slow down the engine a lot, though, in the sense that it needs way more nodes to reach the same search depth. Because now each deviation from the principal variation has to be refuted by a pair of moves, rather than a single one. But this cannot be helped, it is intrinsic in the game. I guess part of it can be blamed on not counting the veto as a turn / ply.

The game is implemented here http://abstractgames.ru
This is a turn-based site where you can play many variations of chess and checkers.

I CAN USE THIS TO NOT PLAY AGAINST THE SCANDINAVIAN!
lol Black can veto your 1.e4

yeah he vetoed nxc1 but then black played nxc1
Great concept, but might be hard to execute in online chess

This is a very creative variant concept. It has a lot of intriguing gameplay aspects.
I find that it has some interesting parallels to Duck Chess strategy. Every move, you block your opponent from making a move that would be advantageous to him. But, with this version, you block a move rather than a square. I think that this could be done online, in a similar way that Duck Chess works. White would move a piece, and then veto-move one of black’s pieces before the real move registers. Black would do likewise.
I have a feeling that chess.com won’t implement this idea (they have a lot of other stuff that they work on), but if they ever did, I’d love to try this. Great idea, Michael!

Duck chess is a sufficiently similar concept that I doubt they would implement this version. It might be interesting to play-test both and see which is better; it's possible they're fairly equivalent to the point that there's no real reason to have both.

reminds me of refusal chess. if your opponent is too distracted you can start eating apart their army with the queen, and refuse *all* queen capturing moves
That’s an excellent variant