Why would my Goodman gas furnace start but not stay on for more than 2-3 minutes?
There are a couple of things that it might be:
It could be the flame sensor. I have seen them where they look clean but are dirty enough that the DC voltage being generated is not enough for the circuit board to keep the gas valve open. I would pull it out and clean it with some scotch brite or a brillow pad. be care full not to damage the porcelain base. It is generally fairly easy to remove, one or maybe two 1/4 screws holding it in place. Gently pull it out and clean it with the pad like you would if you where sanding a wooden dowel.
The 2nd thing it might be (less likely) could be a roll out switch. If your air filter is dirty it would prevent enough air flow across the heat exchanger to keep it from over heating causing the roll out safety switch to open. 
If this is neglected for a long period of time it can cause the ROS to weaken and begin to trip at lower than designed temperature. 
ps. there are typically LED lights on a the circuit board that will blink error codes to help in diagnostics. It is best to capture that info and will usually point you in the right direction.