If your lawnmower is not starting up and you have addressed that its the engine fuel it could be due to bad or stale gas. If stale gas gets into the engine it will cause it not to start or to run very roughly. Best plan is to take it in to be serviced so its done properly.
The easiest thing to do is replace the fuel filter. Drain fuel tank and replace with fresh gas to make sure there is no water or trash in tank. If this doesn't work check fuel solenoid.
First you need to remove all the gas from the tank. then blow fuel lines out, put on a new gas filter. and fill with fresh gas. Let run for a few minutes to get the old gas out of the carb.