The following  Theorem by Nori  (Proposition 3.27 of this [paper][1])  is  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   to take a look at this [survey][2]. There you will find Nori's Theorem 
in Section 2.3.


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**Further comments added later:** 

 1. If $C \subset \mathbb P^2$ is an irreducible curve with only one singularity having 
smooth branches intersecting pairwise transversely then 
the complement of $C$ in $\mathbb P^2$ and well as the complement of its strict transform in the blow-up of $\mathbb P^2$ are abelian ($D = \hat C$,  $E =$ exceptional divisor in the first case; and $D = \hat C$, $E= \emptyset$ in the second).
 2. If you have a reduced connected curve $C = E_1 + \ldots + E_k$ with rational irreducible components,  the  intersection matrix $(E_i\cdot E_j)$ is negative definite, and the dual graph is a tree  then the fundamental group of the complement of a neighborhood of $C$ has been determined by Mumford in this [paper][3].


  [1]: 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.pdf1.
  [2]: https://arxiv.org/abs/math/0510049
  [3]: https://doi.org/10.1007/BF02698717