Well to be honest, you can get away with driving on one that is coolant jacket to cylindar. You certainly don't want to drive one one where oil and coolant are mixing. That'll guarantee rod knock for sure. I drove on a BHG for a while, mostly because I was convinced I didn't really have one, and it was very slight. I pulled apart the engine and it didn't do any additional damage.
To get back more on topic, I didn't notice any change in gas mileage while driving on a BHG. But then again mine was very slight. Logically, there would be a drop in mpg because you're losing compression and therefore the car is not as effecient.