This is not an answer, but it's a bit too long for a comment.

If I'm not mistaken, any group $G$ which embeds densely into a compact Hausdorff group $H$ has the property that it is finite if and only if it has finitely many irreducible representations. This includes the <a href="http://en.wikipedia.org/wiki/Residually_finite_group">residually finite</a> groups, which embed densely into their profinite completions. To see this, use the fact that by Peter-Weyl $L^2(H)$ decomposes as a direct sum $\bigoplus_i n_i V_i$ where $V_i$ are the finite-dimensional unitary irreducible representations of $H$ and $n_i = \dim V_i$. It follows that there are infinitely many $V_i$ if and only if $H$ is infinite if and only if $G$ is infinite; moreover, by density the $V_i$ are irreducible representations of $G$ any two which are inequivalent as $H$-representations are inequivalent as $G$-representations. 

So to find a counterexample we should look for groups that are not residually finite. Infinite simple groups certainly have this property, but I don't know any of them well enough to analyze their irreducible representations.