We assume that vertex weights can be arbitrary positive integers, or more precisely, they can be positive integers at most 2n. Then the current task cannot be performed even in a slightly weaker time bound O(n2), unless the transitive closure of an arbitrary directed graph can be computed in O(n2) time, where n denotes the number of vertices. (Note that an O(n2)-time algorithm for the transitive closure will be a breakthrough.) This is the contrapositive of the following claim:
Claim. If the current task can be performed in time O(n2), the transitive closure of an arbitrary directed graph given as its adjacency matrix can be computed in O(n2) time (assuming some reasonable computational model).
Proof. As a preprocessing, we compute the strongly connected component decomposition of the given directed graph G in time O(n2) to obtain a DAG G′. Note that if we can compute the transitive closure of G′, we can reconstruct the transitive closure of G.
Now assign the weight 2i to each vertex i of the DAG G′ and use the algorithm for the current problem. Then the binary representation of the sum assigned to each vertex i describes exactly the set of ancestors of i, in other words, we have computed the transitive closure of G′. QED.
The converse of the claim also holds: if you can compute the transitive closure of a given DAG, it is easy to compute the required sums by additional work in time O(n2). Therefore, in theory you can achieve the current task in time O(n2.376) by using the algorithm for the transitive closure based on the Coppersmith-Winograd matrix multiplication algorithm.
Edit: Revision 2 and earlier did not state the assumption about the range of vertex weights explicitly. Per Vognsen pointed out in a comment that this implicit assumption may not be reasonable (thanks!), and I agree. Even if arbitrary weights are not needed in applications, I guess that this answer might rule out some approaches by the following line of reasoning: “If this approach worked, it would give an algorithm for the arbitrary weights, which is ruled out unless the transitive closure can be computed in time O(n2).”
Edit: Revision 4 and earlier stated the direction of edges incorrectly.