I have a Tecumseh LV195EA on a Toro # 20016 and when it gets hot won't start. Sometimes it shuts off all by itself. Changed the plug and oil...same.
I change the air filter, plug and oil regularly. I've had lots of trouble finding parts and info on problem. I can fix it, if I know where to start. The plug was a little fouled and the spark plug boot is splitting.
I would suspect a potential ignition module failing at higher temps. To check this, I'd invest in a 'gap type' spark tester. They're inexpensive and available at most auto parts stores. When the unit dies, put the tester in line with the spark plug wire and and see if you still have spark. If not, then you need to verify it's not the kill switch circuit from preventing spark vs the ignition module itself being bad. Basically, you'll want to disconnect the kill switch wiring so that you know it isn't causing the lack of spark. Given the symptoms you describe, I doubt it's kill switch wiring related. It's important to rule this out though so you don't buy a new ignition module and find out that's not the real problem .
So give that a go and let us know how you make out and we can proceed from there. Good luck!
Save yourself the expense of the spark tester, remove the plug wire stick a screwdriver in the plug wire cap, hang on to the metal part of the screwdriver, crank over the engine, if you get a little jolt you have spark, if not you don't, pretty simple. Have a good one. Geo