I am not so used to thinking about integral Hodge structures, so this question might be completely trivial.
What are easy and interesting examples of smooth projective connected varieties $X$ with torsion in their integral Hodge structure?
The Hodge structures arising from abelian varieties have torsion coming from the torsion points on the abelian variety.
I am looking for varieties with torsion in their integral Hodge structure which can't be explained (directly) via abelian varieties (somehow).
The integral cohomology of an abelian variety is torsion-free, so I'm not sure what you mean about them having torsion in their integral Hodge structure.
There are examples of smooth projective surfaces having torsion in their integral $H^2$. (The examples I know best arise as Shimura varieties, but there are examples in other contexts too, e.g. Enriques surfaces. See here for more discussion.)
There will be lots of higher dimensional examples too, but it can be hard to find them discussed explicitly in the literature.