According to Daniel Cassidy’s lunatic work of fake linguistics, How The Irish Invented Slang, the word bounce, as in to throw someone out of a bar, and bouncer, a doorman, derive from the Irish phrase bain as, which Cassidy defines as ‘to extract out of; to remove from, to eject, to extract from’. As we have said before, Cassidy frequently ignored the logical and obvious origin of words and went off on a wild goose-chase looking for some bizarre speculative Irish origin. That is exactly what Cassidy did here.
He says that the OED gives the word bounce as origin unknown. While I don’t have a full copy of the OED handy, I found a pocket OED online which suggests a connection with German bunsen or Dutch bons, both meaning to beat or thump. What is not in any doubt is that bounce is an old word in English, dating back to the 13th century. Furthermore, bain as might have these meanings but would be used of a tooth or something like that. An Irish speaker would talk about throwing somebody out (duine a chaitheamh amach as an áit).
From the Irish side, the claim is inherently improbable. When we add the fact that bounce is an ancient word in English and that its meaning is entirely appropriate in the context, Cassidy’s claim is revealed as the barking mad nonsense it really is.