The fan, who came from the home end, jumped on to the park at the end of the first half and made his way towards the player.
After kicking the ball away from Tavernier, the supporter then appeared to strike out before being hauled away by police.
This comes just six days after Celtic winger Scott Sinclair had a bottle thrown at him during the Scottish Cup match at Easter Road on Saturday.
Having apologised to Sinclair for that incident, Dempster was furious she’d be having to do the same again so soon after with another opposing player.
She told BT Sport: “It’s absolute beyond comprehension that we’ve had another event where we’ve seen this idiotic behaviour.
“I was going to call him a supporter, but I’m not going to call him a supporter. It’s unacceptable. Completely unacceptable.
“This week alone I’ve had to write to Scott Sinclair to apologise for the bottle and now I’m going to have to go see Mr Tavernier and apologise as well.
“It reflects badly on the club, it reflects badly on the Scottish game at a time where the Scottish game is thriving and doing lots of good things. But what are we talking about again tonight? A foolish idiot coming on to a pitch again. It’s completely and utterly unacceptable.
She added: “I missed the entire second half. What should we be talking about? A good game of football on a Friday night. It’s ridiculous.
“This week, after everything has happened, why would somebody come on to the pitch and do that? It’s unacceptable at any point, but given anything that’s going on... words fail me.”