Edge removal to convert a non-planar DAG to a planar DAG while maintaining reachability?
Yes, there's a nice algorithm for this.  Compute the transitive reduction, then check whether the result is planar.

Why does this work?  The transitive reduction is the graph with the smallest number of edges, that has the same reachability relationships as the original graph.  It is also unique.  Moreover, any other graph with the same reachability relationships must include all the edges in the transitive reduction (plus possibly some others).  Thus, if the transitive reduction isn't planar, no other graph with the same reachability relationship will be planar (as adding more edges to a non-planar graph can never make it planar).