I found this theory in comments over at HistoryPorn. After some research, I've only found a few articles of note - one from The Guardian with little sources, one from this webpage which bases the theory on his tuning, as well as another article disputing the theory.
For an example, here is the standard release Cross Road Blues, and here is a slowed version.
There is no documentation indicating that his recordings were intentionally sped up, if that's what you're asking. But then, there exists very little contemporary documentation about Robert Johnson at all!
In the 1930s, audio recording was still in a relatively primitive state, and virtually all recordings were, essentially, documentary. You found the right placement for the microphone(s) and recorded an event as it happened. There was no overdubbing, no tape echo - nothing fancy. Although it is possible that speed of a recording could be altered at the time, it would have been applied specifically as an effect, to the whole track.
Robert Johnson was a relatively unknown musician at the time of his two recording sessions. 'Race records,' as they were known at the time, were steady sellers for the African-American markets but they were by no means blockbusters. Robert Johnson never recorded in a proper studio, either - both of his sessions were recorded in makeshift environments, on portable equipment. In keeping with other 'race records' of the time, Robert Johnson was very likely recorded for 'documentary' purposes rather than intended for unusual production flourishes.
All of Robert Johnson's recordings run at about the same speed. It seems unlikely - though there's no evidence either way - that there would have been a consistent effort to speed up every single recording at the pressing stage, as a sort of post-production decision, to make his recordings more marketable. There are no recordings of him that run 'slowly' - they are all at about the same pitch.
Since he was only ever recorded by one man, Don Law, there is the possibility that Law's equipment was faulty and either recorded him at the wrong speed or played him back at the wrong speed. But if this were the case - and in the Robert Johnson case, people are arguing that there is an extreme difference - you'd think Law might have had his gear repaired between the session in November 1936 and the session in June 1937. Don Law recorded Bob Wills' hit version of New San Antonio Rose in 1938, although it's not clear whether he used the same equipment. Law's recording of Bob Wills is consistent with other recordings of Wills.
Several people who knew Robert Johnson in life have been interviewed, and none of them mention anything about the recordings sounding unusual.
Sorry for the lack of sources - I'm travellng and away from my books.