Claudio Ranieri's men headed into the game knowing a point would move them back to the top of the table ahead of champions Paris Saint-Germain on goal difference.
However, the principality club got off to the worst possible start at the Stade Auguste Delaune, midfielder Antoine Devaux heading in Atila Turan's near-post cross after four minutes.
Monaco responded well, though, and they levelled nine minutes later, Joao Moutinho scoring his first goal for the club as his curling free-kick found the bottom corner of the net.
Reims dictated the play for most of the second half, Floyd Ayite hitting the crossbar as they pressed for a winner.
Turan was sent-off late on for Reims but, despite their numerical advantage, Monaco were left to settle for a share of the spoils.
Nantes claimed victory in the Britanny derby as they leapfrogged neighbours Rennes with a 3-1 win.
Michel Der Zakarian's side took the lead after 17 minutes, Serbian striker Filip Djordjevic tapping in to an empty net after his initial header from Lucas Deaux's cross crashed off the post.
Serge Gakpe doubled the lead three minutes before the interval with a superb right-footed strike, before Portuguese striker Nelson Oliveira pulled one back 10 minutes into second half.
However, Rennes failed to gain momentum following the goal, and Nantes killed the game off with an injury-time strike from Deaux.
Montpellier have now drawn their last five games after they were held 1-1 by Ajaccio.
Jean Fernandez's visitors went ahead through Remy Cabella, the attacking midfielder converting from the penalty spot after Ajaccio defender Claude Goncalves had been sent-off for handling Souleymane Camara's shot.
Montpellier could not press home their advantage, and the hosts earned a point as Ronald Zubar headed in from Johan Cavalli's free-kick nine minutes prior to the interval.
Yassine Jebbour was sent off for a second bookable offence two minutes later, as both teams finished the contest with 10 men.
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