There's no danger in reheating rice that you've refrigerated and as long as the cooked rice has no mold in it or any bad smell to it then the leftover rice is safe to eat and reheat.
Just don't reheat the rice more than once or the rice becomes hard and then it makes it hard to eat but it's safe to eat but you likely would not want to eat it.
Cooked rice will last for up to 6 days in the fridge as long as you keep it properly stored in the refrigerator.
Although the cooked rice will last up to 6 days in the refrigerator it's best to eat the cooked rice within 2 to 3 days so it still has the best quality and freshness to it.
Cooked rice can be left out of the refrigerator at room temperature for at least 2 hours and if the cooked rice has been left out longer than 2 hours then it can have bacteria grown on it and should be thrown out.
The way to tell if cooked rice has gone bad is if the cooked rice has become hard, smells bad or has begun to grow mold or other fungus on the rice.
You can reheat rice a few times but it's best to only reheat the cooked rice once because the rice gets harder and harder and goes bad when you reheat leftover rice too much.