Plants consume oxygen at all times, not just at night. Night is simply the time when the net exchange of oxygen is into the plant; during the day, when the plant can perform photosynthesis, more oxygen is produced than is consumed. Plants need oxygen because plants are made living cells which need to perform cellular respiration to get the energy they need to function.

Admittedly, needing to perform cellular respiration alone does not mean that consuming oxygen is required - many (micro)organisms use fermentation or anaerobic respiration. However, there are no known anaerobic plants; all seem to need to take in oxygen from the environment. Some more information on ScienceLine regarding plants and their respiration.

Plants need energy in order to stay alive. During the day, a plant can get its energy from the sun, but it can't do that at night (unless it's living in a lit up greenhouse, of course). To get energy while in the dark, the plant must consume its energy reserves, which are in the form of sugar, and to extract energy from the sugar, it must use the same oxygen-consuming chemical reaction that you use when you get energy from sugar.

Thus, no, there are no plants that don't consume oxygen when it is dark. A plant that is left in the dark without oxygen would asphyxiate, for exactly the same reason that an animal left without oxygen would asphyxiate. The only difference between plants and animals is that plants can get energy from light, and so don't need oxygen when light is available.

We consume oxygen all day and all night, because we breathe in oxygen and use it to 'burn' our food. That's called 'aerobic respiration.' Without oxygen, you can only burn food part way, and you don't get so much energy from the food - that's called 'anaerobic respiration' and it happens in our muscles when we exercise hard. 'Aerobic' is 'with air.' You can find that stuff here.

It's stressful for plants not to have enough oxygen. Scientists are figuring out the ways plants get energy when they don't have enough oxygen. When plant seeds get flooded with too much water, they can live for a while, but then they will die.

Great question!
Plants, like animals, undergo cellular respiration to break down food (in the form of sugar, or glucose, C₆H₁₂O₆) for energy to live. Respiration requires oxygen to convert the glucose into energy, water, and carbon dioxide. And in fact, plants undergo respiration all the time, both night and day. You may be familiar with photosynthesis, where plants use carbon dioxide (CO₂) from the air, water, and sunlight to create food (again glucose or sugar) and oxygen. But you may not have realized that photosynthesis is really only happening in the green parts of the plant. So while the leaves and stems are producing oxygen during photosynthesis, meaning those cells have available oxygen for respiration, the parts of plants that do not undergo photosynthesis (such as roots), need oxygen for respiration. Thus all plants require oxygen to survive because all plants require respiration to live, and thus there are no examples of plants that do not need oxygen.

In continuation, I suspect you have heard that there are organisms that don't require oxygen to live, and that might prompt you to think there could also be plants that don't require oxygen. What would be more accurate is to say that all plants and animals need oxygen to live, but some microorganisms don't require oxygen to live. You can think of this from an evolution perspective. Earth's atmosphere was not always rich in oxygen and the organisms were anaerobic, meaning oxygen was not required for growth (and in fact oxygen was deadly to these organisms). Then about 2.5 billion years ago, a certain type of cyanobacteria evolved the ability for oxygen photosynthesis. These cyanobacteria started producing oxygen gas which in turn increased the oxygen in the atmosphere. The presence of oxygen caused most of life at the time to die from the toxicity of the newly oxygen-rich environment. This time in Earth's history is often called the Great Oxygenation Event (GOE), and the remaining life after extinction only survived by adapting the ability to live in an oxygen-rich atmosphere.

Plants and animals today come from the survivors that developed oxygenic respiration. Therefore, the only organisms today that can live without oxygen are thought to represent those early, simple structured organisms before the GOE.

That said, scientists are always finding new discoveries. There is a group of tiny marine-sediment dwellers that are considered animals that do not require oxygen to survive. But these complex organisms evolved in an entirely oxygen-free environment, so if there are plants that do not require oxygen, scientists have not found them yet.