Well the fuel pump is in the gas tank. However there are many reasons for a "no start" condition! Remove the air cleaner top, or cover and feed the throttle body (looks like a carb.) a little bit of your lawn mower gas, not much about 2 oz or 1/4 cup is 'plenty'!! Now crank the engine, no throttle needed, to see if it will start. IF it does start and then dies, you may have a fuel pump problem. IF it does not start, you most likely have an ignition problem. Post test results along with the engine 'size' for more infomation and testing YOU can do.
A very definitive test if performed very carefully. But for safety's sake, I just want to suggest a spray can of throttle body cleaner instead of gasoline. Gasoline has no surface tension, allowing it to spread out rapidly on hard surfaces and giving it its volatile behavior. A small amount sloshed outside of the throttle body near an arcing plug wire can turn a vapor cloud to fire in less than a second; I have seen it. The spray type throttle or carb cleaner can be aimed and shot directly into the throttle body. Still, it's a good diagnostic to perform.
Good advice! If it catches fire, the 'ignition' system is OK! Most likely fuel pump problem..... Good idea not to try and pour gas from a 'can' all over the top of the engine! 2oz/1/4 cup poured into this ('down draft') vertical throttle body should do fine. Propane injector is the way to go but few know just what that is!!