Hi, Anonymous your bike starts great when cold but once it reaches operating temperature and the engine is shut off for a few minutes then you try to restart it and it will turn over but will not fire or it doesn't do anything, the usual suspects are:
1. When electronics get hot they build up resistance then stop working. It might be an ignition coil, stator, CDI unit etc. First, you must find out if the spark dies out. When the bike won't start, pull a plug wire off, stick a new spark plug in the end and ground the threads on bare metal while someone cranks the engine over. You should see a nice big blue spark between the spark plug gap. Incorrect spark plug gap will cause or contribute to a no start when hot issue.
2. Fuel Tuners have been known to cause this type of problem, disconnect it and see if resolves the issue.
3. Any engine management sensor could be faulty especially the ones that monitor temperature.
4. Gas cap not venting properly, improperly routed fuel lines they must always go down from the gas tank to the carburetor never in any kind of a loop this will cause a vapor lock. Clogged, bent, kinked, pinched or restricted carburetor vent lines will do the same.
5. Overhead camshaft valves adjusted too tight when cold, this will cause loss of compression when hot, except for pushrod engines they usually get looser when hot.
For more information about your issue and valuable "FREE" downloads that you will need for viewing or printing please click on the blue links below. Good luck and have a wonderful day.
Suzuki DR350 Service Manual