England were left furious after Bairstow was given out stumped when he walked out of his crease at the end of an over, almost immediately after ducking a Cameron Green ball.
The incident looms as a potential turning point in the game, with England going to lunch at day five on 6-243 in pursuit of 371 for victory.
Australian players were booed for 25 minutes after the stumping, and subjected to chants claiming they were "cheating".
Khawaja then took exception to something said by one member and stopped to speak with them, before two stewards stepped in.
David Warner followed him through the long room, and was ushered away by stewards after stopping in front of another fan.
AAP has been told that Lord's members have been reminded of behavioural standards after the incident, but no fans were removed from the venue.
The Lord's long room is one of the most unique experiences in world cricket, where players walk through members.
The Marylebone Cricket Club, owners of the Lord's venue, have been approached for official comment.
