Changing an inner tube in a lawn mower tire is not much different than changing an inner tube on a car or bicycle tire. To change an inner tube without replacing the tire, you don't need to remove the tire completely from the wheel. Removing one side will allow you access that's adequate for inner tube replacement.
Remove the wheel from the lawn mower. How you should do this and what wrenches you will need will depend on your type of lawn mower.
Slip a flat blade screwdriver under the bead of the tire on one side of the wheel. Tires are held on wheels with a bead. A bead is either a thickening of the rubber or a rubber coated cable.
Pry up and out to pull the tire bead over the edge of the wheel.
Slip a second screwdriver next to the first when you have pulled the side of the tire outside the wheel. Slide this second screwdriver around the bead, pulling the tire off the wheel.
Pull the old inner tube out of the tire after you have pulled one side of the tire from the wheel. You may need to push the tube stem into the wheel to get the tube out of the tire.
Slip the new inner tube into the tire. Pay attention to the hole in the wheel for the inner tube's stem. Make sure the stem is very near the wheel's stem hole.
Stuff the tube into the tire. Keep it as flat as possible.
Using your hands, a bit of the tire bead into the wheel. Continue around the wheel until most of the bead is back inside the wheel. If you can, use your hands to finish setting the tire. However, on some tires, the last inch or two of bead may need a screwdriver to pry them in place. If you use a screwdriver, be careful not to puncture the new tube with the screwdriver.
Inflate the tire and re-install the wheel and tire on the lawnmower.