On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. More explicitly, he shows that one can decompose this vector space in terms of weight decompositions and obtain a sort of canonical basis for these differential forms. Is there some literature which does the same for cluster algebras or varieties?
Thomas Lam and I are working on a paper about this, which we hope to get out by next summer. Here 1 2 are some slides describing what we knew last Spring when I talked about these results in Austin. We know more now, but it doesn't look like there is going to be a clean complete answer.
There is not much known in general, as far as I know.
There is a nice 2-form (called the Weil-Petersson 2-form) defined using the cluster algebra structure.
This can be found in article "The Weil-Petersson form on an acyclic cluster variety" by Greg Müller.
Sometimes, the exterior powers of this 2-form provide other algebraic differential forms.