Continuity test buzzer sounding continuously
I'll assume you don't have or can't find a schematic for it (if you do let us know) so:

First thing is to open it up and take a look around the input circuitry, something blown/shorted may be easy to spot visibly (as Jim mentions any protection components are a good bet)
If the rest of the functions are still working, it may be just the continuity circuit which is damaged - if this is the only function which requires the buzzer on the meter, tracing back from the buzzer to any driving circuitry may prove helpful.