This post for linear transformations provides counterexample $(x,y)\mapsto(x+y,y)$, but the projective transformation $[x,y,z]\mapsto[x+y,y,z]$ does have a fixed point $[1,0,0]$ on any fixed line. Is the following true?
Suppose $T$ is an automorphism of $\mathbb RP^2$, for any fixed line $l$ under $T$, there is a fixed point $x\in l$.
If $T$ has 3 distinct fixed points, the lines joining 2 of them are invariant and distinct (no other fixed lines), so any fixed line of $T$ contains a fixed point. If $T$ have only 1 fixed point, I don't know how to proceed.