We were given a cheap push mower when my wife and I moved into our house. It didn't run, and supposedly only ran a handful of times before dying. I decided to get a Honda and see if my brother and I could get it running so he could have it as a backup mower at his house.
So far he has:
replaced and gapped the spark plug
re set the spacing on the ignition coil
replaced the diaphragm/gasket on the carb
replaced the air filter
ran a couple tanks of fuel through it to clear of old gas
Remove the carb, tank assembly, take the carb of the tank, look at the diaphragm just under the pipe where it connects to the intake pipe on the engine. I bet is has been pulled out from under the gasket. the fuel pump is right there and will suck fuel around the diaphragm. Replace with part number 795083 diaphragm.