No, smoke doesn’t usually set off fire sprinklers as their trigger mechanism works off temperature. When the surrounding air reaches a certain temperature range, a chemical reaction causes a liquid in the fire sprinkler to expand and shatter the surrounding glass. This sets off a chain reaction which causes the plug to release the water above, which then flows out and is distributed as a ‘sprinkle’.
The reason they work off temperature and not smoke, like smoke detectors do, is that it’s more important for fire prevention systems and domestic sprinkler systems to avoid false alarms. Smoke is frequently caused by false alarms such as cooking, smoking, and burning food. If the sprinklers were to be triggered by smoke, it would cause unnecessary water damage to the area. By making them run on temperature, this can be avoided. I hope this helps!