The brick on my fireplace hearth seems very dull compared to the shiny refinished wood trim in the den. I would like to put polyurthene on the hearth to give it some sheen. Is it safe to do that?

Safe, yes. Possibly. But it will chip / peel off with heat and time. Instead, clean it with a mild solution of muriatic acid. Available at tile stores. Use 50% acid and 50% water. Brush it on. Use rubber gloves. Protect the work area. Rinse it off when done. You might consider cleaning it with tri sodium phosphate and then rinsing it well with clean water. Thereafter, if it dries too dull, consider using pine sol solution to wash it with. There is a little oil in it and it might just do the trick. And it won't peel later.
Good luck
Mark

Using water based poly on brick is a very nice way to finish the brick. It will brighten up the fireplace brick and make it look great. As a chimney sweep and a former painting contractor, I have seen this used many, many times. Once dry the chemicals in the poly are stable and will not give off any fumes. An open window and a fan in the room during application will handle the fumes.
As far as heat causing problems... you get very little heat on the outside face of a fireplace. If it does cause problems you need to rethink using the fireplace because the smell from the poly will be the least of your worries. The main problem will be your house burning down because the fireplace is not functioning properly.