By Malcev's theorem, every finitely generated linear group is residually finite (RF). On the other hand, say, the group of rational numbers is linear, but is not residually finite. Thus, one has to impose some restrictions on linear groups to avoid this example. Discreteness sounds like a reasonable hypothesis. It is a nice and easy exercise to show that every discrete subgroup of $PSL(2, {\mathbb R})$ is residually finite.

Question 1. (This question came from Fanny Kassel) Are there non-RF discrete subgroups of $PSL(2, {\mathbb C})$?

On the other hand, almost surely, there are no simple discrete infinite subgroups $\Gamma$ of rank 1 Lie groups. (Take a high power of a hyperbolic element in $\Gamma$, then its normal closure in $\Gamma$ should have infinite index in $\Gamma$.) This argument fails however in the higher rank case because of the Margulis' normal subgroups theorem for higher rank lattices.

Question 2. Are there infinite simple discrete subgroups of $SL(n, {\mathbb R})$?

It is hard to imagine that such things could exist, but I see no way to rule them out...