The forgetful functor from group objects in abelian groups to abelian groups (just forget the extra group structure) is an equivalence.

More generally there are lots of "idempotent" constructions, such as "taking abelian group objects in" or "taking pointed objects in" that give examples like this. These are the "uninteresting" ones in that they are very general.

An example not of this type is given by duality data. Let's say you have symmetric monoidal category $C$. Inside there you have dualizable objects, and they form a full subgroupoid $C^{dbl}$. But a dualizable object can also be seen as a bunch of duality data: a tuple $(x,y, x\otimes y \to 1)$ for example. This type of data forms a groupoid $DDat(C)$ and the forgetful functor $DDat(C) \to C^{dbl}$ is an equivalence.

Another example that happens a lot in ordinary category theory is when forgetting some higher dimensional coherence data ends up not being necessary. For example, forgetting from "very structured monoids" to simply monoids is an equivalence, where a "very structured monoid" is a functor $\Delta^{op}\to C$ satisfying the Segal conditions - similarly with "very structured commutative monoids", or more generally algebras or commutative algebras.

Similarly one can forget from very structured monoidal categories to monoidal categories - although that is an equivalence only if one views both as $(2,1)$-categories. But I think the above shows that this phenomenon occurs even in ordinary categorical situations.

Let me give a final example: there are lots of criteria like "to preserve colimits, it suffices to preserve coequalizers and coproducts". This can be seen as a forgetful equivalence: from the category whose objects are cocomplete categories and arrows are colimit preserving functors to the category with the same objects but functors are functors which preserve only certain types of colimits. These statements are easy, but it is still an example of this form.