Adding too much oil to your lawn mowers engine can cause severe engine damage is in blowing the engine apart.
Too much oil pressure builds up in the engine where it cannot lubricate the engine properly and then a piston rod can blow through your engines block causing a hole in the engine block.
I've done it myself before on a push lawnmower and I thought it would be okay but after mowing for about 20 minutes the engine through the piston rod through the engine and threw the oil all out onto the ground.
Luckily the push lawn mower was an older one and I had gotten a lot of use out of it but now I make sure to put the proper amount of oil in the lawn mower and if I happen to overfill it I drain the excess oil out.
You can use a tube and suction bulb to remove the excess oil if you put too much oil in it.