Yes. You can use a high-quality power strip with your fireplace with one very important caveat. The bottom plug needs to absolutely go directly into the wall and should under no circumstances ever be plugged into a surge protector. However, the top power cord can be plugged into a high-quality surge protector, like one used for a computer with a battery backup. This will protect the delicate electronics inside your fireplace.
The bottom cord is for the heater, which draws quite a bit of watts (1500), and should never be plugged into a power strip or surge protector, as it can overheat and possibly cause a fire.
This is very important! NEVER plug the bottom cord into anything other than directly into the wall!