Jon Snow was not permitted to join the feast because it would have been a serious breach in what passes for etiquette in Westeros, to have a bastard son sitting at the same table as the King of the Seven Kingdoms.
I know exactly what instigated Rhaegar to kidnap Lyanna, but I am unsure how much of the story has been told in the TV series. The crown prince's wife took offense to something that someone did/said at a jousting tournament at Harrenhall, and she blamed Lyanna for it.