You could use heaves as the audience is a singular entity and all of the members of it are acting together, so the singular verb form works. BUT that is not relevant if you decide you need to keep it in the past tense to mathc the rest of the sentence and use HEAVED. You certainly should not use heave.
The option to use the present tense there is stylistic and would depend on the broader context of the whole piece.
Either tense works fine with after followed by a gerund. The only rule for after is that when a clause begins with after, the main finite verb(s) in that clause must be in the same tense as the verb of the clause to which the after clause is subordinated:
The audience heave sighs of relief after they watch the acrobat perform. The audience heaved sighs of relief after they watched the acrobat perform.
(By tense here I mean "present" or "past", not to be confused with aspect [perfect, continuous, perfect continuous] or voice [active, "middle", passive].)
And I would only ever use hove for something "heftier" than a sigh (e.g. "He hove his uncle into a hornet's nest.")