Here is just one of probably very many possible strategies for tiling a chessboard that can be easily modified if a white and black (or blue and yellow in this rendering) square is eliminated.
Note that the path shown below is a complete circuit, ending up wherever you start. And it doesn't matter whether you start blue-yellow or yellow-blue, or which direction.
I've picked one blue and one yellow square to eliminate, but which ones clearly doesn't matter. If you start to the right of the missing blue square, you will put down dominoes on yellow-blue pairs along this path ending with the yellow-blue that is next to the missing yellow square. Going the other direction from the missing blue square, you tile the rest of the board, ending up next to the missing yellow square.