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



> **Theorem.** Let $D$ and $E$ be curves on
> a smooth projective 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 $C$ having a triple point with distinct tangents as its only singularity then the fundamental group of the complement 
of its strict transform is abelian (apply Theorem above to $D  = \hat C$ and  $ E = \emptyset$ ). 
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/0510049
  [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