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The following Theorem by Nori (Proposition 3.27 of this paper) 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. There you will find Nori's Theorem in Section 2.3.

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.