No. I'm not sure whether you're asking about pushout along an arbitrary 2-functor, or just about pushing out a locally fully faithful functor along another locally fully faithful functor. Either way the answer is no.
Consider the inclusion of 1-categories into 2-categories as the locally discrete ones. This functor is fully faithful. It has a right adjoint, so it preserves pushouts. And it identifies the faithful functors between 1-categories with the locally fully faithful functors between essentially discrete 2-categories. So it suffices to show that faithful functors of 1-categories are not closed under pushout in $Cat$.
So it suffices to show that injective homomorphisms of monoids are not closed under pushout in $Mon$.
For this, if you're asking about pushout along an arbitrary map, then it suffices to consider the pushout of the injection $\mathbb N \to \mathbb Z$ along the map $\mathbb N \to \mathbb N / (2=1)$, which is the non-injection $\mathbb N / (2=1) \to \ast$.
Otherwise, we're pushing out an injective homomorphism along another injective homomorphism. The result need not be an injective homomorphism -- see here (the first-linked paper at Benjamin Steinberg's answer there gives an example with $\leq 4$-element semigroups; adding disjoint unit elements these are $\leq 5$-element monoids).