After re-reading the question I'd like to amend my answer.
The net definitely gets the benefit of his Saving Throw, for things like the fire ball. Other circumstances may give other results. For example, consider the classic scene: The Hero is tied up, and rolls over to the fire or lamp and holds his wrists near the flame. He's trying to burn the ropes without burning himself. Thus, he's "making his save" while trying really really hard to make sure the ropes don't make theirs.
That being said, the net doesn't necessarily get the benefit of the Protection from Fire either. On that point I'll stick to my original statement, that it depends on whether it's being controlled by someone else, by him, or whether you as the DM consider it an unattended item.
Just as characters can choose to fail a Save, I'd say that the character can choose whether to extend magical Protections *THAT THEY CONTROL* to items they're in contact with.
If the Protection is from a Ring or potion, he has the choice. If it's a spell he cast on himself, or some item he activated, it's his choice. If it was cast by someone else *before the net was thrown on him*, it's his choice. If the net was on him when the spell was cast, it's under the spell.
This is, of course, just my opinion. In the end, it's the DM's opinion that counts.