They can still photosynthesize. Photosynthesis is sometimes divided into the light and dark cycles, but it would be more accurate to call these light-dependent and light-independent reactions. The light-independent cycle, where NADPH is oxidized to NADP+ and CO2 is converted to glucose, can still happen if light is available. It just doesn't depend on photons, like the light-dependent reactions do.
Still, most plants do not do well with all-day light. This isn't because of issues with photosynthesis, but because of a few other factors:
1) Too much light can burn the plant, especially if there is high light intensity or high heat (tomatoes are especially sensitive to this)
2) A lot of flowering plants (including poinsettia) need darkness to flower
3) Plants, like many animals, have something similar to a "circadian rhythm", where periods of light and dark help them regulate things like starch metabolism