My interpretation is that the answer is, "doctrinally, effectively, none." While I will definitely say that it seems like there was a good deal of effect on the politics of Italy (and, to a lesser extent, France) in the High Middle Ages, from a doctrinal perspective, it does not really seem to be terribly significant. And I say this for two particularly important reasons:
The doctrine was already in place:
- The claims and assertions about the primacy of the Papacy had been around since before Eusebius who died in 339.
- Leo the Great is generally considered the major proponent of the doctrine, and died centuries before the document was written.
They were disproven:
- The documents were disproven 70 years before Luther, 100 years before Trent
- While that was not published until the 16th century, it was certainly widely known when the doctrine of infallibility itself was defined at Vatican I
The major challenger to Papal authority in when those were written was the See of Constantinople, but it is apparent that these documents were known only to the west. (And when it was shown in the East, it contributed to the Great Schism)