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.