When looking for a non linear Lie Group I always find the example of the Heisenberg Group $H$ modulo a normal Group $N$. Where the matrix of the two groups are of this form
$$ H = \begin{bmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix} $$ $$ N = \begin{bmatrix} 1 & 0 & n \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
Do you know any other simple example of non-linear Lie Group?
As Travis said, the universal cover of the a special linear group is not a matrix group. Another classical example is the metaplectic group.
Also any nilpotent lie group wich is non simply connected is not a matrix group.
See Wolfgang Ziller lecture notes on lie groups and representations p40 as reference.
Edit: the reduced heissenber group isnt a linear group also