They don't live in the woods. They're SEARCHING the woods for Cinderella, Rapunzel and Sleeping Beauty (oh, those dwarves!). And why shouldn't the princes survive? Not everyone dies in the story. (Sorry, I thought the question was- why do they live in the woods.)
Because one day Angela Lansbury came over to Sondheim's house. And after she mistakenly opened a door she thought was the bathroom and saw Anthony Perkins hanging naked by a leather strap, she sat down with Sondheim. "Steve," she said, "these beautiful princes have to live otherwise people will think they are symbolic of the two princes that died in the Tower of London." Sondheim said, "Ang, I see your point."
If anyone ever tells you that you put too much Parmesan cheese on your pasta, stop talking to them. You don't need that kind of negativity in your life.
Because the show perpetuates the sexist belief that men are entitled to fool around without consequence, but women must pay the ultimate price.
Salve, Regina, Mater misericordiae
Vita, dulcedo, et spes nostra
Salve, Salve Regina
Ad te clamamus exsules filii Eva
Ad te suspiramus, gementes et flentes
O clemens O pia
Well, in the revival, the two princes do not come out with Snow White and Sleeping Beauty during the finale. Their lines are also changed to "The higher the thrown / The longer the fall". So perhaps it could be assumed that they may have met their ends while in search of their new princesses.
Just my guess, but as clever as their duet is, I don't think we care enough about the princes that their deaths would particularly affect us. Even in her few lines, Rapunzel captures our hearts--but if she doesn't, we grieve for her vicariously through the witch.
Killing off one or both princes would merely prolong the plot, without furthering the action.