How long will a trapped animal stay or live in a trap?

Vanilla

Rabbits do not die in traps or player inventory.

Reign of Giants

Rabbits take two days to die in a trap or in your inventory. You can feed most animals food to prolong their lives. Most animals, once dead, will become a meat which will decay into rot if it is not eaten.

You're looking at a great way of stretching out your meat's life-span!

1. Store Rabbits and kill them as they decay
2. Cook any Meat you don't want to consume or dry immediately
   • Cooked Meats last longer than raw Meats
3. Keep all food Dry
   • Soggy food increases rotting speed by 30%.
4. Place any Meat on Drying Rack to extend even further
5. Feed a bird any decayed Meats to get a fresh Egg
6. When using Fruits and Vegetables, be sure to freshen them up by cooking them, if needed.
   • This halves the spoiled amount

Other Notes

Sometimes, an animal such as a mole, which cannot be trapped by a Trap, will trigger the trap and you will come back to an empty, not-working Trap. I mention this because it would look the same as if you let a Rabbit completely rot.

The first answer is half correct. If you are playing Don't Starve without Reign of Giants, the animals will stay alive indefinitely. Reign of Giants added the need to feed live animals that are in your inventory, which also carried over to live animals inside traps. The time is about 3 days until they starve.