Have cemented bridge and pain in tooth - how can I remove my bridge safely?

If there is pain in tooth it is a pulpits.   If it is reversible it will gradually improve, if it is irreversible pulpitis and you will probably need Root canal Therapy.  This can be done by making a small hole through the bridge.  Safest approach.  The best crown remvover is the Kavo Crown remover but I am one of very few dentists who invested in this expensive device that is used only rarely.

You need to see a dentist to find out what's causing your pain. If the cause of pain is the tooth's pulp, root canal treatment could be done without removing the bridge. The risk with attempting to remove a permanently cemented bridge is fracturing teeth in the process. Usually when bridges are removed, they have to be cut into sections, which also destroys them.

You do not state when the bridge was placed.  If it was recently...contact your dentist and ask about the problem.  It may be just a passing sensitivity.  However if you truly classify this as a "toothache"  then advise your dentist immediately.  Removing a cemented bridge can be accomplished but there is always a risk involved.  Gentle put persistent tapping can usually removed it.. though it may take repeated efforts over several visits.  Once accomplished a reassessment of the cause can be determined.  If it is due to decay or a need for root canal therapy..it can be accomplished and the bridge re-cemented as long as the margins are intact.. If not only your dentist can decide if the bridge can still be used.