For the most part I agree with Jan and Stuart, however, there are some unique problems here that differ from a game with a professional dealer:
1. Since the game is "Self-dealt" or "Player Dealt" it would be impossible to retrieve the flipped up card (which should have been the pre-flop burn) without giving an advantage to someone at the table. Who would go through the deck and find that card without gaining an advantage over the other players? Someone has to see the cards in order to find it.
2. Since the 2nd pre-flop burn was also placed into the muck there is no way to identify that card so there is no way one can apply the TDA Four Card Flop rule.
3. The correct or 'best' solution is one that most closely matches as many of the correct cards as possible. If one numbers the 8 cards that constitute the burn and board cards, 1 through 8, Card 1 would be the preflop burn, Cards 2, 3 & 4 would be the flop, Card 5 the burn before the Turn, Card 6 is the Turn, Card 7 is the burn before the River and finally Card 8 is the River.
In this situation, as I understand it, Card 1 is flipped and placed into the muck, Card 2 is erroneously mucked as the "Pre-Flop" Burn, Cards 3, 4 and 5 are now the flop. It is unclear whether there is any "action". In my opinion, in this particular situation, it makes no difference since neither Card 1 or Card 2 can be retrieved without someone involved in the game gaining an advantage (remember the game is self-dealt). So to this point two of the three cards in the flop are 'correct', but card 5 is not. I don't see any possible way to correct the flop at this point, action or not.
Continuing, the closest way to match the remaining board cards is to do exactly what Pokerchip did, rule that Card 2 will be considered the "Pre-Turn" Burn and continue on with Card 6 being the Turn, Card 7 being the "Pre-River" Burn and Card 8 being the River. I believe this solution is the best match to the what should have been correct cards.
One way to alleviate this problem is to have your games place the "burn" cards under the chips in the pot which should be completely separate from the muck. That way the Burn cards can be identified as they are not "in the muck" until after the River is on the board.
Hopes this helps!!