My father has a Troybuilt mower walk behind 4.5 2 years old lawn mower... It only runs when the air filter is removed, shuts off when the air filter is in it. Does anyone have any idea why or what to do?

Sounds like it is running very rich, needing as much air as possible. The filter, may cause it to flood out. Higher fuel/air mixture
I would clean jets, orifices and carb.
Filter not clogged?