There are a few fish and invertebrates that will eat bristle worms. Like with anything live you buy for your tank, it is important to find out how compatible they are with your current livestock and coral.
For example, the Banded Coral shrimp are known to have a voracious appetite for bristle worms. But, they can be aggressive towards fish. Arrow crabs are a good choice for keeping bristle worm populations in check however not all arrow crabs will eat them.
With regards to fish that enjoy them as a meal, Flame Hawkfish and Six Line Wrasse are good options but again, check their compatibility with the rest of your fish, invertebrates and coral.
Wrasses can attack and eat invertebrates if they get too large. I had a wrasse which was great in the 4ft tank. When I got the 5ft tank, she grew ALOT and caught her nipping the tail ends of the two fire shrimp. They were healthy and still alive when she did. We had to unfortunately catch her and move her on.