Is it true that irreducible generic representations of $G_2(F)$ are self-dual?

Since $-1$ is in the Weyl group (over $F$, not just the algebraic closure) you might expect every irreducible representation to be self-dual. This is the case over $\mathbb R$. It is false over a $p$-adic field, but subtle, and it is not easy to construct an example.

There are non-self dual cuspidal unipotent represenations of $G_2(k)$ where $k$ is the (finite) residue field. By the standard pull back and induction procedure these give rise to non-self-dual supercuspidal representations of $G_2(F)$. The same thing works for all exceptional groups.

Dipendra Prasad has a discussion of some closely related matters in A 'relative' local Langlands Correspondence (arXiv:1512:04347).

In terms of involutions, $G_2(F)$ (for example) has an involution $\tau$ such that $\tau(g)$ is conjugate to $g^{-1}$ over the algebraic closure. This is the "Chevalley involution", and it is inner for $G_2(F)$. However $\tau(g)$ cannot always be $G_2(F)$-conjugate to $g^{-1}$ (exactly because then every irreducible representation would be self-dual, which is false.) See The Real Chevalley Involution (arXiv:1203:1901), page 4.

Finally, because of the involution just mentioned, one would expect that every L-packet for $G_2(F)$ is self-dual, but the duality operation could be nontrivial on the packet.

$\DeclareMathOperator\PGSp{PGSp}\DeclareMathOperator\PGL{PGL}$Generic representations of a $p$-adic $G_2$ are indeed self dual. It suffices to prove this for super-cuspidal representations. Observe that, for unitary representations, taking dual is the same as taking complex conjugate. Now all generic super-cuspidal lift one-to-one to generic representations of $\PGSp_6$ by the exceptional theta correspondence, see Savin–Weissman - Dichotomy for generic supercuspidal representations of $G_2$, Compositio Math (2011) (MSN). Since the exceptional theta correspondence commutes with complex conjugation, the statement follows from self-duality on $\PGSp_6$ side. In fact, since any representation of $G_2$ lifts either to $\PGSp_6$ or to a compact form of $\PGL_3$, one can completely classify representations of $G_2$ that are not self dual: the super-cuspidal representations that correspond to non-trivial representations of the compact form of $\PGL_3$.