Every domino covers two adjacent squares, and 31 dominoes cover 62 squares — the arithmetic checks out perfectly. Try it anyway. The proof that you'll always be left holding one impossible domino is a single observation about color, and it generalizes into a lovely theorem about which pairs of missing squares are fixable.
Every domino is a couple: one light square, one dark, no exceptions — even while the board looks gray, each tile you place is quietly holding one of each. The full board has 32 of each color, but the two corners we sawed off were both dark. 32 light + 30 dark means 31 couples can never form: by the time 30 dominoes are down, the two survivors are guaranteed to be light — and light squares never sit next to each other, so the last domino has nowhere to lie.
The fix isn't to try harder — it's to mutilate more carefully. Cut one square of each color and the board becomes tilable again, always.