Brick is quite porous, so the stain is more than likely not coming out. There are stone "stain removers" that are effective with less porous material like granite and marble, where the stain is closer to the surface. You may get some of the discoloration out, but you may also lighten or discolor the brick with the stain removal product. The adhesive will probably damage the brick when you try to remove it. Again, any solvents strong enough to aid in its removal, will probably leave a discoloration behind.

To balance things out, try the removal approach, but then consider a new treatment on the brick. Tinting darker, White-washing, or painting. It all depends on what you'd like for a finished look.

Thanks for the response! I think we'll see what we can remove and go from there. We're waiting to decide on the mantle and overall look until after we figure out what we can do with the bricks.