In a paper called Critical Percolation on any Nonamenable Group Has no Infinite Clusters, Benjamini, Lyons, Peres, and Schramm show that ... critical percolation on any nonamenable group has no infinite clusters.
More precisely, if $G$ is a Cayley of any finitely generated non-amenable group, then $\theta(p_c)=0$. This immediately implies that $p_c(G) < 1$.
The paper also shows that for any invariant bond percolation $P$ on $G$ (invariant means its distribution is unchanged by the action of the group), if in $P$ the expected number of neighbours of the origin $o$ is at least
d_G(o) - \kappa(G),
then with probability one there is percolation. (Here $d_G(o)$ is the number of neighbours of the origin in $G$, and $\kappa(G)$ is the Cheeger constant of $G$, which is positive since the group is non-amenable.) This gives that
p_c(G) \le 1-\kappa(G)/d_G(o)$1-\kappa(G)/d_G(o),
which is weaker than the bound in Vincent Beffara and Asaf Nachmias' answers, but applies to a broader range of percolation models.