So far I have not seen any indication that approval does anything besides flavor. Are there any actual game mechanics tied to how much approval you have with characters?
There is no bar or other mechanic like in the previous Dragon Ages that tell you how far along you are with their approval. For the most part, you have to go by how they address you when you talk to them outside of combat, such as at Skyhold. For example, if Cassandra is your friend but isn't romanced, she'll often start the conversation with 'my friend.'. Or Varric, who will usually open up with a quip or joke of some sort if he's your friend.
There is also a visual representation on the 'cards' of the character when you go to pick your party to head out into the world. While I'm unsure if it's tied to their approval or simply to acknowledge that I have done their personal line of quests, it seems that only the characters that I'm friends with have their cards changed. Take that as you will.
I can't confirm whether it's the case in DA:I, but in previous games (and other Bioware games), the approval rating was directly tied to the Romance options, certain conversation options and personal quests.
Incidentally, I've only been able to do the personal quests for characters I seemed to have a good approval rating with. The same goes for my romance options. But it's hard to check, since I can't see how high the rating is anywhere.
To add to the above points. If someone's approval rating is low enough they will actually leave. This is confirmed to be the case with Blackwall, Dorian and Cole. It does not happen with very story critical characters (Cassandra, Solas etc) but they will hate you and you will get an interesting cutscene (punching Solas, Cassandra getting drunk and telling you off)