I would like to know if there is an explicit example of a finitely presented group that can not be realised as the (topological) fundamental group of a normal complex quasi-projective variety?



Fundamental groups of normal complex quasi-projective varieties
Yes, sure. Take any polycyclic group which is not virtually nilpotent (e.g. 
upper triangular matrices with entries in $GL\_n(\mathbb{Z})$, $n\ge 2$). This cannot be the fundamental group of a normal quasiprojective variety by a theorem of Nori and myself [Solvable fundamental groups of algebraic varieties..., Composito 1999].
There are many other sorts of examples.


