Every functor $\mathbf{Set} \rightarrow \mathbf{Set}$ I can think of preserves monomorphisms (i.e. injective functions), including:
- $\mathrm{Hom}(X,-)$,
- $X \times -$,
- $X \sqcup -$,
- and the constant functors.
The monads I can think of all have this property, too.
What are some natural examples that don't?
Not a real answer on your question, but it might be enlightening.
Every functor preserves sections (i.e. split monomorphims) and in Set almost all monomorphisms are sections.
The only exceptions are the functions $\varnothing\to Y$ where $Y\neq\varnothing$.
These are the only functions that are monic, but do not have a left-inverse.
So it is not so strange that endofunctors in Set that do not preserve monomorphisms are hard to find.
In order to find a counterexample (or maybe a proof that all monomorphisms are preserved by endofunctors in Set) you will have to focus on the mentioned functions.