Is it permitted to warm up a fully-cooked item on Shabbat?

Since it is forbidden to ignite a fire on Shabbat, this question only applies if the flame on your stove is on since before the Shabbat. It is also forbidden to put even a fully-cooked item on the stove on Shabbat.

It is permitted, however, to return a pot or pan onto the fire if it was already there beforehand -- for example, if you took off the pot in order to remove some food1 and now you want to return it to the stove -- provided the following conditions are met:

1. The food-item is fully cooked.
2. The stove-top is covered (see above).
3. Originally, when you took the food off the stove, you intended to return it onto the stove.
4. Your hand did not leave go of the pot handle throughout the time it was off the stove.
5. The food wasn't transferred to a different pot.
6. The food hasn't completely cooled off.

It is however permitted to warm up a fully-cooked food if the food is not situated on top of the flame. For example: you can warm up a Challah by placing it atop the Cholent pot which sitting on the already covered stove-top.

Note: It is forbidden on Shabbat to remove food from a pot or pan which is situated on its heat source. You must always first remove the pot from the blech or hot plate before taking food from the pot.