Gentlemen:
These scenarios can become more complicated when one of the players holding the extra card has acted, or is one of the blinds. The hands with too many cards are certainly dead. What do we do with the blinds?
As far as the original question, with only one player having an extra card, as long as substantial action has occurred the hand will play out to conclusion. In my opinion, I would not burn a card before the flop. This would assure that the proper board will be preserved. The single player with too many cards has a dead hand. Period.
In the case of multiple players with too many cards, there are two different decisions I'd apply: #1) if recognized before substantial action, I'd rule a misdeal...#2) If two or more players have too many cards and they are not yet involved in betting but substantial action has occurred in front of them, their hands are dead and the hand will play to conclusion. #2.1) Because the proper board would be impossible to recreate, I would proceed by burning and turning the remaining deck stub as usual.
There are always unusual situations that can complicate what sometimes appears to have a very simple solution. Always try to protect the proper cards whenever possible. This means that: we must strive to preserve the board that would have tabled if no mistake were made.
In the original situation with one player holding an extra card, pre-flop followed by substantial action, the player with too many cards has a dead hand and the flop will be turned without a burn. In the next situation where two players have too many cards, their hands are dead and the flop would be preceded by a burn...what else could we do?