I was curious as to which surfaces of the hive boxes I should cover with Tung Oil. I’ve seen websites that tell you which surfaces to coat when painting.
It seems to me that all surfaces (including top/bottom edges) should be coated with oil except the inside of the brood boxes & supers?
Can someone confirm that for me? In particular, I was wondering if I should apply oil to the inside of the roof cover of the FlowHive.
Only the outside. Bees do the inside. There is no harm to doing the inside if you are using Pure Tung Oil, but it isn’t necessary. The top and bottom edges are up to you - they are not exposed and the bees don’t care. I wouldn’t do them - it is too much extra work. 
Thanks! So, should I leave the underside of the roof untreated or should I treat the entire roof?
I would treat the outside underside (the eaves, if you like), but you don’t need to seal the “ceiling” which will be inside the hive. There is no harm in sealing it if you want to, just it is unnecessary.