Suppose that in a KQ vs. K endgame (or in an endgame where Black can't possibly mate the other side), White makes two illegal moves. Can Black claim a win?
No, the game is drawn, as per article 7.5.5 of FIDE Laws of Chess (emphasis mine):
7.5.5 After the action taken under Article 7.5.1, 7.5.2, 7.5.3 or 7.5.4 for the first completed illegal move by a player, the arbiter shall give two minutes extra time to his opponent; for the second completed illegal move by the same player the arbiter shall declare the game lost by this player. However, the game is drawn if the position is such that the opponent cannot checkmate the player’s king by any possible series of legal moves.