Almost certainly the number of animals on the planet would plummet within days. Humans do have ways to synthesize organic nutrients. We could use technology to produce and recycle organic matter for a very long time, but not for a very large population. We also have mechanical and chemical means of scrubbing CO2 and adding oxygen to the atmosphers. The human race (in small numbers) could survive indefinitely, but it is hard to imagine what life would be like.

If all plant life were to disappear, ultimately so would almost everything else -- not from lack of oxygen, since it would take decades for the oxygen already in the atmosphere to deplete -- but from starvation. We rely on energy coming into the system of earth's interlocking biospheres, to power our metabolisms. That energy capture happens with plants, which turn heat and light from the sun into energetic molecules which can be broken down in a way that releases that energy in small steps that can be utilized by our bodies. If there were nothing performing photosynthesis, then those energetic molecules would never be created in the first place. If you look at various food chains, you will find that either organisms eat other organisms that eat plants, or they simply eat plants, or else they *are* plants; the plants are the base that everything else exploits in order to get the starches and carbohydrates that power us and enable us to put vital proteins together. Would the human race survive? Only as long as our canned and preserved food stores held out, unless we had invented Star Trek-level technology which could create food out of constituent molecules...and we're not even close to that yet.