Does engine completely stall (requiring a re-start) or does it come back to life if you let off the throttle?
Ok so first post here, and I know the thread is kinda old, but right now I'm working on a Tecumseh that does exactly what you said, and you never posted a solution in this thread, so I made an account to ask.
A little info: Yerf Dog 3203 kart somewhere around 2001-2002. It was wrecked a while back, and then went to some guy and got some work done... Then sat for a while... Then went to another guy to get running, then sat for a while again. Yada yada yada. The whole bulb on the carb where you pump the gas to get started was just in tatters when i got to it and barely hanging on. It runs pretty decent with it messed up, but it still runs like crap and has the problem you mentioned above.
Right now, there's just a piece of duct-tape over where it used to be. The bulb was so crappy it was pointless to leave it on there.
So, is this bulb my problem? or is it something else? The motor idles and stays alive, but the idle is rough sometimes. The carb probably needs to be cleaned, but I can do that myself with the help of the internet.
Anyhow, if you see this post and respond with useful information, thanks in advnace!