Plants stop doing photosynthesis at night due to absence of sunlight, not because they are tired. If a plant is provided strong enough light 24x7, it will continue the process of photosynthesis nonstop. The plant will continue to grow 24x7 under light in short term; however it will eventually affect the plant in an adverse manner, long term. The effects will vary from species to species.
Artificial light at night is often provided to lab grown & green house plants for faster plant growth & to skip the seasonal impact on growth phases, so that many generations of plants can be grown within short time period.
Plants produce their own food in the leaves during photosynthesis at day time. This food is continuously transported all over plant body. This process of food transportation is slower than the speed of food preparation. So, this rest phase at night time helps to overcome this lag of food transportation speed, when no extra food is produced.
As far as flowering is concerned, under 24x7 light ‘Long day plants’ will tend to flower. ‘Short day plants’ will stop flowering. ‘Day neutral plants’ will not have any effect.
Increased light period will help the plants to grow much faster, but 24x7 light will eventually affect the plant growth in adverse manner depending on the species.