I have a 14.5 HP Intek OHV model 287707 that started blowing gray smoke out of the exhaust after running a couple minutes, then it dies but starts right up again. Also, the oil smells like gas. No excessive oil leaks. Seems to run fine when first started up. Could this be a carburetor problem?

Yes it is a carburetor problem. The float and needle are stuck open and it is flooding fuel into the crankcase. That is why your oil smells like gas and you have gray smoke. I will bet that it is hard to determine the exact oil level on the dipstick.

First, do not run the engine until corrected or you risk engine damage. The fuel in the oil acts like a thinner destroying the lubrication properties of the oil.

Second, take the carburetor off and clean/rebuild it. When you are satisfied with the carburetor, including the operation of the float and needle, reinstall.

Third, change the oil, for the reason stated in the first.

Fourth, with fuel in tank, note level to make sure not changing by draining through the carburetor.

Fifth, remove the spark plug and turn the engine over a few times to get oil back to the surfaces that need lubricated on startup. Reinstall spark plug

Sixth, check fuel tank level for drop, if no drop, Try starting the engine.

I hope this helps you get it going.