I guess some of the discussion above is relevant here but in a nutshell, if your tooth has died, or is dying then you either need root canal treatment to save it, or an extraction.
Thus, the signs and symptoms you need a root canal
are basically the same as for an extraction, so click the link to read about them. A common exception is for teeth that are seriously loose and painful from gum disease. In these circumstances, a root canal won't achieve anything, and extraction is commonly the only option. Where teeth have broken off under the gum, you should also brace yourself for having the tooth removed.