Since it's a home game, the rules are those agreed by the players. Anyway it's not probably a good idea to return the burned cards to the deck, and just proceed to deal the next card.
Casino dealers are trained to follow very specific procedures to prevent this kind of thing, but mistakes do happen. A floorman should always be called if an error occurs, and he will make whatever ruling is fairest to the players, with guidance from some standard principles:
Once significant betting action takes place after exposing a card, it must stand. That is, if an incorrect turn card (for example) is shown, and there's a bet and call before this is discovered, the card must stay and play continue.
If a card is exposed before betting action is complete on the previous round (for example, the last player on the flop has not called and the turn card is exposed), that card cannot play. If it was the turn the dealer should turn the card that would have been the river, then reshuffle the stub and continue.
If the error is discovered before betting action, the card should be replaced by the correct one if possible. If this is not possible, then play should be continued in a fair manner determined by the floorman.
As always, the floorman has discretion to take any other action he deems necessary to protect the game.
In a home game, substitute "floorman" with "player consensus".