When your chainsaw quits after it gets hot, you either have a problem with gas flow being restricted into the engine or with the spark. This is a common occurrence on older-model chainsaws. By replacing your spark plugs and keeping your gas lines clear you can reduce the chances of this problem recurring.
Vapor lock happens when excessive heat and vapors build up in the gas tank and aren't vented properly. Check for this problem the next time your saw dies when it's hot. Unscrew the gas cap, vent the gasses and try to restart your saw. If it starts you have vapor lock, and will need to carefully inspect the fuel tank's vent hole to see if it is plugged or damaged. Clean this hole with a brush and rag.
Another common problem is an old or dirty spark plug. When the plug gets covered in black carbon the spark will have difficulty firing, especially when hot. You may also have a problem in the ignition module or the coil in the module. Again, run the chainsaw until it quits and then squirt a little starter fluid into the open choke valve, underneath the air filter. If it starts up you likely have a fuel system problem, but if it doesn't start, your ignition module or coil is likely almost finished.
Check the Fuel Supply
At certain times when the engine gets hot, the flow of fuel to the carburetor can get restricted. This can occur with a dirty fuel filter on the end of your main hose. The hose itself, when hot, can also lose its proper seal to the carburetor, restricting the flow. Replace your fuel filter and hoses on a regular basis, usually once a season. Try adjusting your carburetor's screws to see if you can richen the fuel mixture in the carburetor.
A dirty or worn-out carburetor can also cause similar problems when hot. You should get your carburetor cleaned and serviced. Try installing a carb kit to older carburetors to help with the flow of fuel. A fuel pump inside the carburetor can also start failing and any warping in the diaphragms will also cause similar problems. You can check for a dirty carburetor by squirting a shot of carb cleaner into the open choke hole just after starting the engine. If you see a lot of thick, dark smoke pouring out the muffler, your carburetor is dirty.