Legia took their case to the Swiss-based court after their punishment for fielding an ineligible player was rejected by UEFA's appeal body on Thursday.
But CAS confirmed on Monday afternoon that it had denied Legia's initial request and that Celtic will play Maribor in Slovenia on Wednesday night.
Legia will now travel to Kazakhstan to face Aktobe in the Europa League but they will continue their legal battle against UEFA, which ruled their 2-0 second-leg win over Celtic be forfeited because they had field a player that they mistakenly believed had served a suspension.
A CAS statement read: "The Court of Arbitration for Sport (CAS) has denied the urgent request for provisional measures filed by the Legia Warszawa SA in the course of its appeal against the UEFA Appeals Body decision of 13 August 2014.
"The UEFA Appeals Body had confirmed the result 3-0 (forfeit) in favour of Celtic FC against Legia in the second leg of the third qualifying round of the UEFA Champions League. The UEFA considered that Legia had fielded an ineligible player (Bartosz Bereszynski) during the match.
"Legia Warszawa filed an appeal to the CAS at 17:30 on 15 August 2014 together with an urgent application for provisional measures requesting that it be provisionally admitted to participate in UEFA Champions League matches until the CAS renders its final arbitral award.
"The appeal is directed against UEFA and Celtic FC which were invited to comment on the Polish club's request until 13:00 (Swiss time) today.
"The President of the CAS Appeals Arbitration Division considered the parties' written submissions and has denied Legia Warszawa's request. Accordingly, Celtic FC remains qualified to participate in the UEFA Champion's League 2014/2015 and will play its next match against NK Maribor on 20 August 2014.
"The CAS arbitration will now continue and a panel of arbitrators will be constituted to examine this case within the next weeks. On the merits, Legia has requested that no sanctions be imposed on either the club or the player Bereszynski and that UEFA be ordered to pay a financial compensation to the Polish club."
The punishment, which turned a 6-1 aggregate win for the Polish side into an away-goals defeat, stemmed from Legia's failure to register Bereszynski in their squad list for the second qualifying round against St Patrick's.
That meant his three-match suspension had not officially been served when he came on as an 86th-minute substitute as Legia beat Celtic 2-0 at Murrayfield.