From what I understand (never went and grabbed one!), they don't "bite" and they don't really sting. Their "bristles" are what are the issue - if you brush up against them they'll break off and imbed in your skin. It's kind of like working with fiberglass insulation all day and having that itchy feeling... but worse.
With that said though, if I was allergic to bees etc... I wouldn't think about going into my tank without gloves. There are plenty of things in there other than bristleworms that can cause you problems. Lots of nasty bacteria in there to start with, and if you have corals... well... that's a whole different story. For most folks it's not an issue, but if you have any type of allergies to bees or any venomous insects/animals, I'd be very very careful around your tank.