The following  Theorem by [Nori][2] (Proposition 3.27)  is very much in the spirit of what you are looking for.



> **Theorem.** Let $D$ and $E$ be curves on
> a non singular surface X. Assume that 
> 
>  1. the only singularities of $D$ are nodes;
>  2. $D$ and $E$ intersect transversally;
>  3. every irreducible component $C$ of $D$ satisfies $C^2 > 2 r(C)$ where
> $r(C)$ is the number of singularities of  $C$.
> 
> Then the kernel of the natural
> morphism $$ \pi_1(X-D-E)
> \longrightarrow \pi_1(X-E) $$ is
> abelian.

In particular if you take an **irreducible** plane curve having a triple point with distinct tangents as its only singularity then the fundamental group of the complement 
of its strict transform is abelian. As Dmitri pointed out, if you drop irreducibility 
this is no longer true. 


You might want also to take a look at this [survey][1]. There you will find Nori's Theorem 
in Section 2.3.


  [1]: http://arxiv.org/abs/math/05100491.abelian.
  [2]: http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1983_4_16_2/ASENS_1983_4_16_2_305_0/ASENS_1983_4_16_2_305_0.pdf