UEFA announced the punishments, which are subject to appeal, on Tuesday after finding fans of both clubs guilty of racist behaviour and other infringements.
Zenit face last year's beaten finalists Borussia Dortmund in the knockout stages, with the first leg taking place in Russia on February 25.
Olympiacos' opening meeting with Manchester United takes place in Greece on the same evening.
Both clubs have three days to appeal their respective penalties, which include fines. Zenit have been ordered to pay €40,000 and Olympiacos €30,000.
A statement on UEFA's official website explained the reasons for the sanctions, which relate to incidents that occurred in Olympiacos' game against Anderlecht on December 10 and Zenit's clash with Austria Vienna the following day.
It read: "Olympiacos have been sanctioned for the racist conduct of supporters (Article 14 of UEFA Disciplinary Regulations), insufficient organisation (Article 33 (4) of UEFA Safety and Security Regulations), the setting-off of fireworks (Article 16 DR) and the use of a laser (Article 16 DR).
"There will be a partial closure of the Greek club's stadium – the lower tier of the north stand – for their next UEFA competition home match. They have also been fined €30,000.
"Zenit have been penalised for the racist behaviour of fans – namely the displaying of a banner (Article 14 DR) – the setting-off and throwing of fireworks (Article 16 (2) DR) and crowd disturbances (Article 16 DR).
"There will be a partial closure of the Russian team's stadium – specifically the area housing the club's ultras – for their next UEFA competition home game. Zenit have also been fined €40,000.
"Both clubs may lodge an appeal against the respective decisions within three days of receipt of the written verdict."
