Former Manchester City youngster Mee sustained the problem during Sunday's 1-1 derby draw with Blackburn at Turf Moor.
Mee completed the 90 minutes at left-back but took to his personal Twitter page on Tuesday afternoon to reveal the bad news of an expected four-to-six week lay-off.
He tweeted: "Managed to tear my medial ligament in my knee midway through the game on Sunday, looking like a 4 to 6 week job #gutted".
A club statement from Burnley later confirmed the news, concurring with Mee's prognosis that he would be absent for "several weeks".
Clarets boss Sean Dyche said: "It's a blow for Ben and we will have to give the knee a chance to settle down and then fully assess the injury.
"We are hoping it will be sooner, rather than later, but we are expecting it will be weeks, rather than days."