Heidi, there is probably a layer of sealer or a finish like polyurethane on the brick. Water combined with the sealer and made the hazy white ring. The sealer will probably like getting rid of the water and replacing it with oil if you give it the oil to work with. Daub on petroleum jelly (Vaseline), or mayonaisse, or any thick liquid with a lot of oil in it. You want it to stay there for at least over night and work. Things like motor oil and WD-40 will not work for this as they don't stay around.
If this doesn't work, come back and we can tell you about more drastic measures. but it probably will. Use your coasters or plastic plant pot holders.