What engine is it??? It could be anything from electrical to a stopped up idle circuit to a damage cylinder causing high reistance. So many possibilities.
I doubt if it is the fuel pump as it will run if the throttle is held up, BUT might be flooding the engine due to a hole in the diaphram but they ususally will not pump fuel if ruptured. Also there is enough fuel in the carb bowls that will allow it to run for 30 seconds to 2 or 3 minutes depending on the engine so the fuel pump would not be needed to run for that short of time.
I can walk you through some ignition tests if you have a timing light or some old plugs and jumper wires. Feel free to call me tomorrow Later in the afternoon as I just came up from the garage and going to get some sleep!!