When you instructor said to "run it dry," he meant for you, at the end of your flying day, to pull the fuel line and let the engine run all of the fuel out of it. You don't want an alcohol based fuel sitting in your engine. it will absorb water and can lead to rusting your bearings.

The other mention of "running it dry," was to not turn a brand new, never run, engine over a lot. You are just wearing the piston by doing this. An ABC type engine needs fuel and lubrication, as well as heat from combustion to allow the cylinder sleeve to open up to running clearance. This type of engine is supposed to have a pinch at the top when cold. When new, you like a "crunchy" feel. Tighter is better.