The plants would ultimately die.
Although the plants won't need oxygen to make photosynthesis occur, it would be unable to perform cellular respiration. They will be unable to respire and produce the product of cellular respiration which is CO2 + H2O + Energy. These are necessary reactants for photosynthesis. Eventually, photosynthesis will consume more CO2 than it produces with cellular respiration. They won't have a sufficient food source and eventually die off.