Does there exist a noncompact connected Lie group with a finite-dimensional, unitary, faithful, irreducible representation over $\mathbb{C}$?
If you remove any of these hypotheses except that of finite-dimensionality (unitary, faithful, irreducible), I believe even $\mathbb{R}$ will produce a counterexample. If you remove the hypothesis of finite-dimensionality, I believe $SL(2,\mathbb{R})$ will produce a counterexample. But what if you assume all of the hypotheses?
A classic issue!
Of course, the basic answer is that such things tend not to happen... Nevertheless:
The first point is that the full unitary group of a finite-dimensional Hilbert space ("Hilbert", to make sense of the unitariness) is compact. So the purported group $G$ would continuously inject to a compact group. And/but to assert that a non-compact group admits a continuous homomorphism to a compact group does not instantly yield a contradiction, if one keeps in mind an irrational winding of $\mathbb R$ in a (compact!) two-torus.
To disallow the latter pseudo-example, one might add hypotheses, ... semi-simplicity excluding abelian-ness? Depending on reactions, one can add things here...