As I know all cells require oxygen. So my question is how efficiently can plants operate in no-oxygen atmosphere? Do all plants produce enough oxygen for themselves? Can they consume their own oxygen or the oxygen will be quickly lost to the air? Is there a plant that consumes more oxygen than they produce?
During the daylight, the plant is photosynthesising faster than it is respiring so there is no net uptake of oxygen (the oxygen of course being produced in the as part of the photosynthesis).
Of course, this only applies for tissues where photosynthesis is occurring. In the roots of the plant, oxygen must always be present in the surrounding soil/growth medium for respiration. Therefore if there is no oxygen in the roots then the root cells will be unable to produce ATP from respiration and consequently die. This will eventually lead to the death of the entire plant as it is unable to take up nutrients due to the lack of a functioning root network.
This is the cause of plant death when soil is waterlogged - the water fills spaces in the soil that would otherwise contain air (i.e. oxygen).
Certain plants (often crop plants), rice being given as an example in the comments below, are able to survive being waterlogged and the resulting low oxygen supply. However they can only do so in the short term. Rice plants in particular have adaptations that allow the transport of oxygen from the aerial parts of the plant down to the roots. This does of course mean that the rice can not survive being totally submerged for an extended period of time. Plants such as rice may have better adapted anaerobic fermentation pathways that become the main producer of ATP when aerobic respiration has arrested thus allowing them to better cope when transiently submerged than other plants.