It comes just a week after Celtic's Scott Sinclair had a glass bottle thrown at him while taking a corner at the same stadium.
Tavernier tried to grab the ball to take a throw-in but a Hibs yob kicked it away from the defender after he jumped over the ad hoardings and run towards him.
The fan then got involved in a melee with the Gers skipper after he appeared to have shoved the player in the face.
Security guards quickly got involved in the ugly scuffle and dragged the troublemaker away from the 27-year-old right-back.
And he was seen then sticking his tongue out as he was being escorted by the security.
But Rangers hero Ally McCoist did not hold his tongue back and slammed the "imbecile" during the half-time break.
He said: "'Make no mistake about it – that's a complete and utter imbecile.
"With any luck he'll be spending at least tonight in a cell.
The bizarre incident comes just a week after Celtic's Scott Sinclair had a glass bottle of Buckfast and a coin thrown at him as he went to take a corner at the same arena.
Hibernian drew with Rangers 1-1 after Florian Kamberi and Daniel Candeias both got on the scoresheet and Hibs defender Darren McGregor was shown a red card in injury-time.
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