You need to observe the furnace for the 5 minute run time. Let us know if both the burners AND main blower stop at the same time. If that's the case I would not expect a high limit safety to be the cause, depending on how the furnace is wired. But the flame sensor associated with the pilot flame could be a problem.
You need to observe the pilot. First, it is a standing pilot... flame always on even when the furnace is not on. Or is it an intermittent pilot that uses a spark ignition system. You may have a flame sensor problem where the pilot flame is not being seen by the control board and shutting everything off. Usually the flame sensor is part of the pilot assembly. That's why it's important to observe how your pilot system works.