It can be a common procedure to perform a root canal on a tooth that is part of a bridge, without removing the bridge. After the tooth is anesthetized, a hole can be made through the bridge to access the inside of the tooth. Then the root canal procedure can be done. After the root canal treatment is completed, the access hole can be filled with a restorative material.
There are some circumstances where the bridge has to be replaced:
If there is porcelain on the bridge, it may fracture when the access hole is made, and a new bridge will have to be fabricated.
Another situation that may occur is the bridge may have a cavity under it (a potential reason for needing a root canal) and to repair and access the cavity, the bridge will have to be cut off and remade.