It's certainly fine to keep them in your closet as long as it stays cool inside, since your running a heater. I don't think the heater is going to make it warm enough to ruin your fragrances, but it's best to be safe than sorry.
You will find some members actually keep there expensive frags in an electric wine cooler or mini fridge! I keep mine out on my dresser because my bedroom does not get that warm and my fragrances are out of direct sunlight and they do just fine.