If a group $ G $ admits a faithful irreducible finite dimensional complex representation $ \pi: G \to GL_n(\mathbb{C}) $ then the algebra generated by $ \pi(G) $ is all of $ M_n(\mathbb{C}) $. So in particular the center of $ G $ must be contained in the center of the full matrix algebra, which is the diagonal copy of $ \mathbb{C}^* $ , $ \mathbb{C}^* I $. See
If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(V)$
If, moreover, $ G $ admits a faithful irreducible finite dimensional unitary representation (which is equivalent to the original hypothesis together with $ G $ being compact) then the center of $ G $ must be contained in the diagonal copy of $ U_1 $.
What happens if we drop the hypothesis of finite dimensionality? If a Lie group $ G $ admits a faithful unitary irrep can we still conclude that the center of $ G $ is isomorphic to a closed subgroup of $ U_1 $?