I just found this when searching around the beatlesbible.com web site:
Birthday was a straightforward song to record, and all four Beatles contributed. Backing vocals were added by Pattie Harrison and Yoko Ono, and all - including the group's assistant Mal Evans - recorded handclaps.
The effects heard towards the song's end, and in the 'I would like you to dance' section, were created by a piano microphone fed through a guitar amplifier with effects added.
Based around a standard blues structure, Birthday contains one of McCartney's finest vocal performances on the White Album. Perhaps the soundtrack to The Girl Can't Help It - which featured Little Richard, Gene Vincent, Eddie Cochran, Fats Domino and others - played a part.
So......who knows.