I have recently read about induced representations and I have the following perhaps naive question about them.
Let $G$ be a finite or infinite (Lie) group. Can we construct all irreducible unitary representations of $G$ (except perhaps for 1-dimensional ones) via induction from some of its proper subgroups?
No. For finite groups, the basic problem (as observed in the comments) is that a representation induced from a subgroup $H$ of a finite group $G$ has dimension divisible by $[G : H]$. In particular, if $H$ is a proper subgroup then an induced representation can't have dimension $1$, but there also exist finite groups $G$ such that the smallest index of a proper subgroup is larger than the smallest dimension of an irreducible representation which is not $1$-dimensional.
For example, take $G = A_5$. Since $A_5$ is simple, any homomorphism out of $A_5$ is either injective or trivial. In particular, the smallest $n$ such that there is a nontrivial homomorphism $A_5 \to S_n$ is $n = 5$, from which it follows that $5$ is the smallest possible index of a proper subgroup of $A_5$. But $A_5$ has irreducible representations of dimensions $3$ and $4$.
However, see Brauer's theorem on induced characters.
For Lie groups the problem is much worse: depending on what you mean by induced representations, almost all nontrivial induced representations are infinite-dimensional. But there are more sophisticated things one might mean by induction that can fix this; see the Borel-Weil theorem, which can be interpreted as a kind of "cohomological induction" (from a Borel subgroup).