Is the isomorphism problem between posets represented by DAGs GI-complete? Given two directed acyclic graphs, how hard is the problem of checking whether the partial orders they represent are isomorphic? Is this problem GI-complete?

I believe this problem is equivalent to asking whether their transitive closures (or their transitive reductions) are isomorphic.
Graph isomorphism is GI-complete for DAGs: https://en.wikipedia.org/wiki/Graph_isomorphism_problem#Complexity_class_GI. 

The problem for partial orders is also GI-complete: We can reduce bipartite graph isomorphism (which is GI-complete) to 2 instances of DAG isomorphism where the DAG equals its transitive closure by considering two canonical ways to turn a bipartite graph into a DAG, that is, either directing all edges from partition 1 to partition 2 or directing all edges from partition 2 to partition 1.