The general principle is to try to preserve as much of the original board as possible, where possible, in these situations. However, if action has occurred based on the wrong board cards, the plays must stand.
I'm sure someone on the Board will step in to correct any errors I may have made below:
1. The dealer burns and turns the flop cards (or the turn or river card) before the previous rounds betting is complete.
Premature flop: Complete action, leave existing burn, flop cards get shuffled back into deck, cut, deal new flop without burning. Continue as normal.
Premature turn: Complete action, leave existing burns (2), burn and turn new card (what would have been river) in place of existing turn. Continue action. Old turn card gets shuffled back into deck, cut, deal river without burning.
Premature river: Complete action, leave existing burns (3), river card gets shuffled back into deck, cut, deal new river without burning.
(Variations: There are some TDs who take a more strict approach with the premature flop situations, by dealing out what would have been future burns and board cards face down before reshuffling, to preserve those board cards. For example, before reshuffling flop cards in the deck, one might deal out what would have been the correct second burn, turn, third burn, & river cards face down onto their expected positions on the table, and then shuffle to redeal the flop. This preserves the correct turn and river cards, which can be flipped up later if necessary. However, this practice is not standard, not great for game security, and under random card theory, probably unnecessary. Another alternative is to always burn a card whenever one is dealing new flops, turns & rivers, even if it means ending up with 4+ burn cards. This is better for game security, and I believe is enforced in some EP rule sets - however, this practice has not really caught on on a broader scale. Some also propose to reshuffle the incorrect turn back into the deck without dealing the "proper" river in place of the turn; this also has not been widely followed.)2. the dealer fails to turn over a burn card before the flop (or turn or river) cards are dealt and the error is spotted before action is taken.
No burn before flop: If it is possible to know which card should have been the burn, use that as the burn card, and complete the flop with the next card. If it is not possible to know which card should have been the burn, scramble the flop cards and choose one to be the first burn card, complete the flop with the next card. Continue as normal.
(Variations: some TDs will simply apply four-card rule here using the next card from the deck. However, this means that 1/4 of the time, you will get the same 'incorrect' flop. But the rule is more straightforward to apply and I think there is some value in keeping less cards exposed. I have also seen some TDs return the flop cards into the stub and reshuffle, but this seems a bit drastic when we know that at least two of the three flop cards are correct.)No burn before turn: Use the turn card as the second burn card, deal new turn, continue as normal.
No burn before river: Use the river card as the third burn card, deal new river, continue as normal.
3. same as Q2 but the error is noticed after action has been taken.
After the flop: Complete flop action. Burn twice (#1, #2). Deal turn card. Continue as normal.
After the turn: Complete turn action. Burn twice (#2, #3). Deal river card.
After the river: Complete river action with existing river card. The exception is if both players were all-in before the river, in which case you would correct the error (i.e. use the river card as the third burn card, and deal the proper river).
4. What is the definition of action for Q3, ie is a fold action or just a bet?
Any action on the board, including check, means that the existing cards must stand. Does not necessarily have to be "substantial action". You probably wouldn't run into the issue of whether a fold is action for this purpose, since presumably he'd only be folding when facing a bet.
Variation: Some TDs interpret 'action' as needing to be substantial action. That is, presumably, "check-check" facing an incorrect board would be ignored, and board cards may still be redealt in those cases. I personally do not agree with this as it may give the players who haven't acted an unfair advantage over the players who have already checked. 5. The dealer burns two or more cards. Again the error is noticed either before or after action has occurred.
Two burns before flop: If it is possible to know which cards should have been the proper burns, use the second burn as a flop card, use the flop card that should have been a burn card as the second burn card, continue action, deal turn card without burning, continue as normal. If it is not possible to know which cards were the proper burns, take the second burn card and use for the flop, scramble the flop and draw one card to use as the second burn card. Continue action, and then deal turn card without burning, continue as normal.
(Variations: some TDs will simply apply four-card rule here using the second burn card and the flop. However, this means that 1/4 of the time, you will get the same 'incorrect' flop. But the rule is more straightforward to apply and I think there is some value in keeping less cards exposed. I have also seen some TDs return the flop cards and one or both of the burn cards into the stub and reshuffle, but this seems a bit drastic when we know that at least two of the three flop cards are correct.)Two burns before turn: Use the burn card that should have been the turn card as the turn card, use the old turn card as the third burn card. Continue action, deal river card without burning.
Two burns before river: Use the burn card that should have been the river card as the river card, old river card is not used. Continue action.