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 quasiprojective variety?

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. 


More obstructions for this realizability problem can be found in the paper On the fundamental groups of normal varieties by Donu Arapura, Alexandru Dimca, and Richard Hain (http://arxiv.org/abs/1412.1483). For an explicit example, take the rightangled Artin group corresponding to a path on 4 vertices, $$ G=\langle a,b,c,d \mid [a,b]=[b,c]=[c,d]=1 \rangle. $$ It is known from http://arxiv.org/abs/0902.1250 that this group (or, for that matter, any RAAG whose associated graph is not a complete multipartite graph) is not the fundamental group of a smooth complex quasiprojective variety. In the paper quoted above, the authors show that such a RAAG cannot be realized as the fundamental group of a normal complex quasiprojective variety. 

