Kazan had been hopeful that the pitch at their Tsentrainiy Stadium would be fit for the 11 April match, but a UEFA inspection found that conditions would not be suitable.
The Russian League club knocked out Atletico Madrid and Levante to get to the last eight, with both of their home legs taking place at the Luzhniki, which has an artificial pitch.
Chelsea have bad memories of the Luzhniki, having lost the 2008 Champions League final on penalties to Manchester United at the Moscow venue.
A statement on Chelsea's website said: "Following a pitch inspection in Kazan, UEFA are of the opinion the pitch and climate conditions at the Tsentrainiy Stadium, Rubin Kazan's home ground, cannot guarantee the best possible conditions."
The first leg of the quarter-final will take place at Stamford Bridge on 4 April.
Meanwhile, Chelsea's Premier League away game against Fulham, postponed due to their progress in the FA Cup, will now take place at Craven Cottage on Wednesday 17 April.