If the door is left open for any length of time with my fridge the whole thing stops. I think it is to stop frosting and acts as a warning that the temperature is wrong. Shut the door and it starts up within a few minutes. Could still be the bulb though.
my position is the same as stanleyman I think it is something to do with the temperature because the door did not fully close, after the door is properly sealed after a while the light operates ok.
This sounds more like the door switch is getting stuck in the closed position. Somewhere there will be a switch that the door touches when it closes and this puts the light off. Find it and check if the light goes on/off when you push it by hand, if it is staying in the off position maybe something sticky has been spilt in/on/around it ?