If your lawn mower starts up and then the engine shuts off within a few seconds of it starting then it's likely that the carburetor is clogged or the pickup tubes in the gas tank that attach to the carburetor could be clogged.
You could also have moisture or water droplets in your fuel tank. Water droplets can also cause your lawn mower engine to stop within a few seconds after your lawn mower engine starts.
Check the fuel and drain the tank into a clean jar and see if you see water droplets. If so dry out the tank and then put some fresh fuel in it.
You should also clean your carburetor out as well to remove water droplets.
If your gas tank doesn't have water droplets in it then you may need the carburetor cleaned by taking it off and soaking it in some parts cleaner such as B 12 Chemtool.
Another thing if gasoline was left in the tank over the winter then the gas will turn to varnish and that could be what is causing your fuel pickup tubes or carburetor to be clogged.
The varnished gasoline will be sticky and can cause a lot of trouble and it will require your fuel tank to be cleaned as well as your carburetor.