Tilings

Our chessboard would not be a chessboard if its cells were not colored black and white alternatingly. As it turns out, this coloring is crucial in answering the question at hand.

Notice that, regardless of where it is placed, a domino will cover one black and one white square of the board. Therefore, $31$ dominoes will cover $31$ black squares and $31$ white squares. However, the board has $32$ black squares and $30$ white squares in all, so a tiling does not exist.

We know the diagonal of the chess board is all the same colour. So we removed 2 of the same colours by removing the opposite corners, so now there is 30 of one colour and 32 of the other. A domino covers one of each colour and so after we using 30 dominos we have two of one colour left over which cannot be covered by a single domino ( because it covers 1 of each colour not two of the same).