It occurred during Slavia Prague's win at relegation-threatened FK Pribram.
The FACR said fourth official Marek Pilny and additional assistant referee Jiri Jech would lose their licences and face disciplinary action.
A statement said Pilny was "visibly drunk", adding that Jech had allegedly "urinated during the game".
Pilny's actions were caught on video and posted on YouTube.
The statement added that Pilny "struggled for balance" and "ran with the linesman copying his movements".
FACR head Miroslav Pelta said "they were not tipsy, they were drunk" and "they don't exist for us any more".
Pelta also said the game's referee and two linesmen would be suspended until the end of the season because they failed to prevent the situation.
Two match officials who appeared to be drunk during a top-flight game have been banned by the Czech Football Association (FACR).