Yes, there are examples where $V$ is a variety of algebras and the left adjoint to the forgetful functor $U: V \to \mathbf{Set}$ is not injective on isomorphism classes of objects. Here are some.

* Take the algebraic theory consisting of no operations and the single equation $x = y$. Then $V$ is the category of sets with at most one element, and $U$ is the inclusion. The left adjoint $F$ maps the empty set to the empty set and every nonempty set to $1$.

* Take the algebraic theory consisting of a single constant $c$ and the equation $x = c$. Then $V$ is the terminal category, and $U$ maps its object to the one-element set $1$. The left adjoint $F$ maps everything to $1$.

If an algebraic theory has the property that there is at least one algebra with at least two elements, then for each set $S$, the unit map $S \to UF(S)$ is injective. This is Exercise 2.3.11 in my book *Basic Category Theory* (though I'm sure people knew this long before it appeared there). There are only two theories that don't satisfy this condition, and they're the two above. But right now I don't see how it relates to the condition that $F$ is injective on isomorphism classes of objects.