What happens to complexity classes if all $\#P$ problems have polynomial-time algorithms? As title says what happens to other complexity classes if all $\#P$ (Sharp-P) problems have polynomial-time algorithms? What happens to PSPACE?
By definition, $\mathbb{P} = \mathbb{NP}$, since any $\mathbb{NP}$ problem could be solved by answering the question "Is the # of accepting paths non-zero", which by assumption, can be calculated in $\mathbb{P}$. As a consequence, the polynomial hierarchy collapses as well.

You also get other results by using results or their contrapositives.