I use a pair of medium priced, bent handled scissors that I reserve just for cutting felt, and I sharpen them using a Fiskars® scissor sharpener. It looks like the one in the photo.
The blades of the scissors slide into the opening on the right, and then are closed and opened as if cutting something. Do this a few times and dull scissors are ready to cut again!
There are several sharpeners on the market. Take a trip to your local fabric store to look at your options.