The needle may have varnish on it making it stick, especially if the vehicle was sitting for a long period of time with old gas in it. You didn't bother to list the model, engine size, type of carburetor, or any other details, so I can't tell which type of needle and seat you have. Some of them use an "N"-shaped piece of wire between the tab on the float and the needle to help pull it open. Others rely on fuel pressure to push the needle off the seat.
I'd start by replacing the needle and seat. You may have to buy a carburetor rebuild kit to get them but those kits aren't real expensive. If it still won't open, remove the float, lightly push the needle closed, then see it fuel pressure pushes it open when you crank the engine. If it does so poorly or intermittently, check for leaks in the fuel supply line that lets air get drawn in. If the engine runs okay for a few miles before this problem starts, it's almost certain the pickup screen inside the gas tank is plugged or collapsing. Replacements are very inexpensive. The clue is the engine will start and run okay again after waiting a few minutes, but each run will be shorter and shorter before the problem occurs again.
Saturday, June 28th, 2014 AT 10:41 PM