The defending champions were looking to close the gap on Paris Saint-Germain, but they had to settle for a stalemate to put them eight points behind the league leaders.
European-chasing Montpellier are six points adrift of a top-five spot, meanwhile, having won just two of their last nine league matches.
There was to be no way through over the 90 minutes for either side, despite racking up 29 goal attempts between them.
In the end the sides had to settle for a share of the spoils, however, giving PSG a chance to move closer to the title when they take on Nantes on Sunday evening.
MONTPELLIER HSC (5-4-1): Lecomte; Aguilar, Mukiele, Mendes, Hilton, Roussillon; Skhiri, Piriz (Lasne 82'), Sambia, Ikone (Mbenza 65'); Sio (Dolly 65')
MONACO (4-2-3-1): Subasic; Toure, Glik, Jemerson (Raggi 46'), Jorge; Fabinho, N'Doram; Lopes (Jovetic 64'), Lemar, Ghezzal; Keita (Diakhaby 71')