There are many halachic issues concerning leaving food on the fire from before Shabbat. The use of a blech or Shabbat hotplate alleviates these concerns. Hazara is a general prohibition of returning or placing food on the fire on Shabbat itself. It is forbidden under all circumstances to put any type of food on an open flame on Shabbat. This applies even to hot, solid, fully cooked foods.
One might ask, "Why is it prohibited to put even cooked food on an open fire? If it’s already been cooked before Shabbat, it can’t be cooked a second time? The reason is that when one puts it back on an open flame, even though it’s already cooked, it appears that it is now being cooked for the first time. This concern is known as Mechzay K’Mivashel.
Therefore, the Rabbis said, if one uses a blech or a Shabbat hotplate, there is no concern of Mechzay K’Mivashel. Since these apparatuses are not the normal method of cooking, it does not appear that one is actually cooking.
It would then be permitted to reheat cooked food on Shabbat, under three conditions:
First, the fire must not be an open flame. One must use either a blech or a Shabbat hotplate to avoid the issue of Mechzay K’Mivashel and the concern that one may adjust the heat (stoke the coals).
Second, the food must be already cooked. If it hasn’t been cooked, one would be violating the Torah prohibition of cooking on Shabbat.
Third, we must determine whether the food must still be hot. If it’s a liquid food, it must still be hot. Reheating a cold liquid constitutes an act of cooking and is forbidden under any circumstance.
However, if the food is solid, like rice for example, there are poskim that permit taking it even from the refrigerator, on Shabbat, and putting it back on the blech or hotplate.
This is Hacham Ovadia’s opinion in Hazon Ovadia as well as in Yehave Da’at. This is also the position of the Menuchat Ahava (1:3, Rabbi Moshe Halevi (1961-2001)) and Hacham David, in his new book (p. 247).
This is also the opinion of Hacham Yitzchak Sternbaum in Yitzchak Yiranen. He explains that one is allowed to put cold solid food from the refrigerator back onto the blech or hotplate even though the Shulhan Aruch rules that one is allowed to return something only if it wasn’t put down yet. Having a blech or hotplate allows one to take it out of the refrigerator, which is considered "put down", since this is a different apparatus and not the usual method for cooking.
Summary: It is always forbidden to put food on an open flame on Shabbat. It is always forbidden to reheat cold liquids, even on a blech or hotplate. One can be lenient and take cold, solid food from the refrigerator on Shabbat and put it on the blech or hotplate.