**EDIT:** As René Pannekoek points out in a comment, I misread the question. This question was asking for *birational automorphisms* of $\mathbb{P}^2$, not just *automorphisms*.

I'll leave this answer (which doesn't answer the question) since it might be useful.

**A negative result for elliptic curves:** Not in general. For example, suppose that $C$ and $D$ are the same elliptic curve inside $\mathbb{P}^2$. Set $P \in C$ to be an inflection point, ie a point such that $L \cap C = 3P$ for some line $L \in \mathbb{P}^2$. Fix $Q$ to be another point which is not of that form (ie, such that the tangent line intersects $Q$ at another point of $C = D$). In particular, let's say that $Q$ is an element of infinite order.

Suppose that $f : C \to D$ to be a map which sends $P$ to $Q$ (note that such maps always exist since $C$ is an elliptic curve and thus an Abelian variety). No automorphism of $\mathbb{P}^2$ (ie, an element of $PGL(2)$) will restrict to $f$, since such an $F$ will send the line $L$ going through $P$ to a line going through $Q$ (and only $Q$).

**EDIT:** As François Brunault points out, this sort of construction can't work to avoid *birational automorphisms*.

**A positive result for elliptic curves:** On the other hand, if your elliptic curve is $C = V(y^2 - x(x-1)(x-\lambda) )$, with inflection point at infinity $P$, then Exercise 4.3 in Chapter IV of Hartshorne says that every isomorphism of $C$ that leaves $P$ fixed comes from a linear automorphism of $\mathbb{P}^2$.

**A positive result for genus 3 curves:** The result is true for genus 3 curves, see Hartshorne, Chapter IV, Exercise 5.7(a).

**A positive result vacuously:** Of course, ``most'' curves of higher genus have no automorphisms at all (ie, most elements of the moduli space have no automorphisms except the identity). See for example Baily, *On the automorphism group of a generic curve of genus >2*.