Let $\Pi$ be some counting problem which is known to be #$P$-Complete.
No. Counting independent sets in graph is #P-hard, even for 4-regular graphs but Dror Weitz gave a PTAS for counting independent sets of $d$-regular graphs for any $d\leq5$ . (In the model he writes about, counting independent sets corresponds to taking $\lambda=1$.)
Computing the permanent of a 0-1 matrix is also #P-hard (this is in Valiant's original #P paper ) but, relaxing your requirement a little, there's an FPRAS due to Jerrum and Sinclair . This corresponds to counting perfect matchings in bipartite graphs.
 Mark Jerrum and Alistair Sinclair, "Approximating the permanent." SIAM Journal on Computing, 18(6):1149–1178, 1989. (PDF)
 Leslie Valiant, "The complexity of computing the permanent." Theoretical Computer Science, 8:189–201, 1979. (PDF)
 Dror Weitz, "Counting independent sets up to the tree threshold." STOC 2006. (Unpublished full version: PDF.)
Adding another example I came across, with an even stronger result:
A (deterministic) FPTAS exists for the problem of counting the number of matchings in a bounded-degree bipartite graph, while this is a $\#P$-complete problem.