Hard to know what the most cringeworthy line in III is, but for me the cheesy "Hi Vincent I remember you" line that Mary says is pretty awful. And the "acting" made it even worse.
And another thing I never understood in the "plot." Why was Altobello out to get Michael? He starts out as an old family friend from Vito's time (of course there is no mention of a Don Altobello in I or II) at the end of his life, who contributes money to the Corleone Foundation. Apparently he and Zasa and Luchese are in cahoots somehow, I think, yet when Altobello introduces Vincent to Luchese he says Vincent is the "hero" who killed Joey Zasa, and Luchese says "we don't want another Joe," so maybe he wasn't sponsoring Joe. IF you buy into the theory that Luchese wa using Altobello to stop Michael from taking over Immobiliare, how would Luchese know Altobello had such animus toward Michael that he would betray him? Talk about lazy. Someone needs to find out who did the continuity in this picture.