If the ignition refuses to even try to crank there's either a failsafe preventing it or an electrical fault. Modern John Deere lawn tractors come with a failsafe that prevents the engine from cranking if no one is sitting in the driver's seat or if the emergency brake is disengaged, so make sure neither of these is the issue. Lift the hood of the tractor and look at the leads connected to the battery. First, ensure that the leads are securely attached, the black wire is connected to the negative terminal and the red wire is connected to the positive terminal. Check the fluid level of the batteries. If it's low, top it off with tap water to the level of the split rings. Charge the battery for several hours before trying to start it again. While you wait, examine the fuse box. If there is the smell of burnt plastic or any signs of charring inside the box it's a sign that you've blown a fuse and need to replace it. Your engine should crank now.
Engine Refuses to Crank
Engine Cranks but Won't Start
First, make sure the throttle is in neutral and the choke is completely open in the starting position. Make sure the gas tank is full and there is fuel moving into the engine. If the engine won't start there's either an issue with the spark plug or the air filter. Remove the air filter and clean it or replace it depending on how dirty it is. Unscrew the spark plug and look it over for gunk or corrosion. A little solvent should be sufficient to clean it, though you may have to manually adjust the distance between the two contacts. If it's too dirty or more than a year old you might want to consider replacing it altogether. The engine should start. If it doesn't ,this is indicative of more serious engine problems, which should be addressed by a professional repair service.
An engine will smoke if its crank case is overfilled with oil. An excess amount of oil will increase the internal oil pressure of the engine block and cause unnecessary stress and wear. Siphon out a quart of oil and see if it is still smoking in an hour's time. If it is, this is because air is getting caught in the engine block, causing the oil to smolder. The thing to understand is that an engine block is a closed system. Any air inside is siphoned out by a device called an engine breather every time the engine starts. There is, however, one spot where air can get in if you're not careful: the dipstick. If the dipstick is broken or improperly seated, it leaves an entrance for outside air once the engine starts. Check your dipstick. If that isn't the problem, then the engine breather itself is broken or faulty. This is a difficult part to replace, though easy to get to, as incorrectly removing or installing a new one simply leaves a second opening for air to enter and makes the problem worse.