If all the photosynthetic organisms on earth die now, how long can human beings survive?
That is a very debatable question. The answer is you can't know! A lot of countries have developed artificial environments secretly which will become operational if such a thing happens, and the capacity of such set ups is unknown to the general public so estimating any value would be futile. Other than that I don't think so humans will survive for more than an hour. There are other organisms of course which can switch to other modes of respiration if oxygen is not available.
If it happens the human being an animals wouldn't exist and microscopic living things would be left. In the case of plants, they obtain their energy directly from sun. But in the case of animals they do this indirectly. If there was no photosynthesis, plants and animals likely could not exist.
That time the atmosphere would have very little oxygen because photosynthesis releases a large amount of oxygen into the air which has been ceased. It is possible that some very basic organisms could still exist like the deep sea vent bacteria that don't get their energy from the sun.
The survival of human beings will depend on the amount of oxygen would be left after the end of photosynthetic organisms.
Humans can still survive for at least a year or two on mushrooms that they can grow on dead organic matter.
In the meantime they can start culture of chaemosynthetic bacteria brought form hydrothermal vents on ocean beds, bring out new & hybrid strains & start its culture. Once they get this success, various filterfeeding animals can be cultured on which humans can survive endlessly.
I guess very short, if even.
To my knowledge, photochemical organisms, like the humans, are a fundamental part of the special conditions that make possible the presence of (our) life on planet earth.
(Although it remains unjustifiable and unexplicable, the devastating imbalance caused to these conditions, just by the human species. Probably it is a degenerative epidemic of insanity).