Find every edge for which every s,t-path in a DAG goes through that edge
Compute the dominator tree of the flow graph $(G,s)$, i.e., the graph $G$ with source node $s$.  Let $s=v_0,v_1,\dots,v_{k-1},v_k=t$ be the sequence of nodes in the path from $s$ to $t$ in the dominator tree.  Check each pair $(v_i,v_{i+1})$ of consecutive nodes in this sequence; if it is an edge of $G$, it is visited on every $s,t$-path, so output it.  It is possible to see that this enumerates all such edges.
The running time is nearly linear time.  It is possible to compute the dominator tree in nearly linear time, and then the rest of the algorithm clearly runs in linear time.  In particular, the Lengauer-Tarjan algorithm computes the dominator tree in $O(E \alpha(E,V))$ time, where $\alpha(E,V$) is an inverse Ackerman's function -- thus nearly linear.  There is also an $O(V+E)$ time algorithm by Buchsbaum, Kaplan, Rogers, and Westbrook and by Georgiadis and Tarjan, but it is more complex to implement in practice.  If you are interested in ease of implementation, you might also be interested in work by Cooper, Harvey, and Kennedy.