Lee put up a valiant fight but was beaten by China's Lin Dan.
The Kota Alam Shah state assemblyman, M Manoharan, an ethnic Indian, said he "deeply regretted" his tweets.
"I sincerely hope my apology will be accepted," he said.
Manoharan had taken to Twitter after the Olympic Games badminton final on Sunday night, sarcastically calling the shuttler "Lee Chong AWAY" and suggesting that the premier should give monetary rewards promised for Lee to "poor boy" Lin Dan.
"If Chong played better than Lin Dan, why Chong Wei did not win? So who is talking nonsense?" he tweeted, to outrage from the local twitter sphere who pointed out that Lee had been injured and therefore had less time to train.
