A lawn mower will run and operate without an air filter but you really should not use the lawn mower without an air filter because the debris in the air such as dust, grass etc can and will get into the carburetor and intake of the lawn mower which can cause engine problems.
Air filters for law mowers are pretty cheap and best to have an air filter on the lawn mower when using it.
I admit though that I've used a push lawn mower before without an air filter because I couldn't get to the store and get a new air filter right away and needed to mow the lawn because the grass was getting out of control.
The lawn mower is still running fine after that but I would not recommend it.