Maxime Barthelme was sent off for the home side late on at the Stade du Moustoir, but neither team could find a winning goal.
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Lorient had a great chance to break the deadlock in the 18th minute, but Vincent Aboubakar's header from a corner came back off the crossbar.
The home side went close again just before the break after hitting Reims on the counter, but Raphael Guerreiro curled his shot just wide from the edge of the box.
Lorient were not far away from a goal 10 minutes into the second half, but Alain Traore's long-range effort flew just wide of the target.
Reims thought that they had found an opener just short of the hour mark, but Guerreiro brilliantly cleared a header off the line from a corner.
The hosts then instantly hit back, when a scramble in the Reims penalty area resulted in Aboubakar chipping against the bar for the second time in the match.
Lorient were reduced to 10 men with 19 minutes left to play when Barthelme was shown a straight red card for a two-footed lunge.
The stalemate keeps Lorient two points clear of the relegation zone in 17th, while Reims climb to fifth ahead of Sunday's fixtures.