In this Signpost interview, Graham87, who uses the JAWS screen reader, reports that the only way to read diffs is by selecting "view source" so that the class names will be announced. This seems awkward.
It seems to me that we could easily make a diff formatter which prefixes insertions, deletions and changes with appropriate messages.
Consider, as a possible model, the way legislative amendments are styled, e.g. http://www.legislation.nsw.gov.au/bills/160508ed-8c69-4e16-ab27-7f2eeaf5f778 (pdf). This is a format which is intended to be understandable when read aloud.
We would need the diff formatter itself, and also a way to select which formatter you wanted to use, say via a link on the diff page.