Erschler has shown that there exists a central extension $G$ of 
$\mathbb{Z} \mathbin{wr} \mathbb{Z}$ by a finite group $F$ which
is not residually finite. Thus the short exact sequence 

$1 \to F \to G \to \mathbb{Z} \mathbin{wr} \mathbb{Z} \to 1$

provides an example of a non-linear group which is an extension of two linear groups.

A. Erschler, Not residually finite groups of intermediate growth,
commensurability and non-geometricity, J. Alg. 272
(2004), 154--172.