When it will not start is the tractor smoking, if no it is most likely a fuel problem.
The fuel tank is not venting, at the back of the fuel tank behind the SMV sign there is a vent make sure it is clean and the tube is clean or temporally just open the fuel tank cap.
Junk in the tank, drain some fuel from the valve at the bottom, back left hand side of the tank.
Clean out the priming pump as the fuel runs up through there here is a video of someone that could not start their tractor for that reason.
JD 5203 John Deere tractor fuel filter change - YouTube
As already suggested summer fuel conditioner JD part number is ty26784.
Make sure the air filters are clean.
Here are some URLs that you may find informative.
//manuals.deere.com/omview/OMRE71736_19/
Leaking Fuel Cap : Solve a leaky fuel cap on some older 5000 series tractors
Loose Muffler on Some 5000 Tractors : Quick fix for loose mufflers or exhaust leaks on certain 5000 series tractors.
//www.deere.com/en_US/ag/servicesupport/tips/augusta/5000/instrpanelfog.html
//www.deere.com/en_US/ag/servicesupport/tips/augusta/5000/bentcylinderrods.html
//www.deere.com/en_US/ag/servicesupport/tips/augusta/5000/backlash.html
This is a John Deere manual on the engine.
//www.visaevolution.com/dati_sito/manuali/officina/motori/john%20deere/ctm125-3029-ENG.pdf