Photosynthesis in oxygen free enviornment
If that "environment" is a closed vessel or bell-jar; certainly the plant would survive in light (when it simultaneously perform respiration and photosynthesis), and in darkness (when it can perform only the respiration) it would survive upto certain time due to the oxygen it accumulated. But should die at prolonged darkness when the plant would finish all the oxygen.

(as suggested by Priestley's classic 1870 experiment * there was enough oxygen to keep a mouse for certain time)



(image link) 



However, if we look the effect of various concentration of O2 on photosynthesis rate without altering the CO2 concentration; yes the photosynthesis rate get affected; one known effect is called Warburg effect or Inhibition of photosynthesis by O2.

In this effect, photosynthesis rate only decrease if O2 concentration is increased. And in vice-versa, with O2 concentration decrease, photosynthesis rate only increase. (source: this, this, this and this)

This take place due to mainly 2 causes: 1. O2 work in  competitive way with CO2 for binding with the enzyme RuBisCO, the key enzyme for CO2 fixation in plants. 2. And that induces photorespiration (in C3 plants). (Wikipedia shows reference to here). (However this old paper also tells other hypotheses)



There are 2 type of responses in photosynthesis rate in at different O2 concentration in various plant species.

One group of plants ( known to bel C3 plants, those do not have any any CO2 concentrating mechanism) ), is remarkably affected by O2 concentration; i.e. affected by Warburg effect in true sense.

Another group of plants ( known to be C4 plants, those have a certain type of CO2 concentrating mechanism); does not significantly get affected by O2. I.e. doesn't show the Warburg effect.
(see also : table).



By the way, It seems photosynthesis would increase than normal  if we could make the environment "completely oxygen-free" but other parameters same. (due to elimination of O2inhibition) But I could not find any report about such "completely oxygen-free" conditions; and seemingly it is not possible because while doing photosynthesis; some  of the evolving oxygen molecules will pass through the cytoplasm; some will move in the intercellular space, and few molecules will stay diffused in atmosphere around the plant. We can only decrease the concentration of O2



Also; Warburg effect is completely about photosynthesis rate. It is not about plant's overall health. Removal of O2 increase photosynthesis rate... it does NOT mean the plant would live healthiest in oxygen deprived condition 




  Sources:
  
  
  Fundamentals of Plant Physiology by Dr. V. K. Jain; revised edition; S. Chand Publications, New Delhi. 
  * Biochemistry/ Lubert Stryer/ 4th Edition/ W. H. Freeman and Co.