Retrieving the shortest path of a dynamic graph
The problem you are asking for is a well-known algorithmic problems. It is actually still open, how hard this problem exactly is. Also you should know that there are different incarnations of this problem. In contrast what you are asking for, usually only the distances are returned, whereas you are asking for the the actual shortest paths. Notice that these paths can be already very long. Dynamic graph algorithms distinguish between edge deletions only (decremental dg algorithms), edge insertions only (incremental dg algorithms) and edge insertions and deletions (fully-dynamic dg algorithms). Thus you are interested in incremental algorithms.

The algorithms mentioned in the post of AJed, are slightly outdated. There are newer results by Thorup, see here, on page 8 for a short survey. The currently best fully-dynamic exact APSP algorithm by Thorup (for distance queries, not the path), needs $O(n^2(\log n +\log^2 (1+m/n))$ amortized update time, while supporting $O(1)$ worst case query time. Notice, that if you have $O(n\log n)$ edges, then you could just recompute from scratch with Dijkstra and Fibonacci-heaps and get the same running time as in Thorup's algorithm. So if your graphs are not to dense, I would recommend using Dijkstra.

I am not aware of any better incremental algorithm. But you should do a web search if there exist newer results for this specialized problem.