It was shown by Alan Turing that (in a certain precise sense) the only connected Lie groups approximable by finite groups are the compact abelian Lie groups, i.e. $U(1)^n$. See Theorem 2 of This paper.
If you allow infinite discrete subgroups then an approximable connected Lie group must be nilpotent and further any simply-connected nilpotent Lie group is approximable. See This paper.