I was using an app (Flappy Bird) recently, and as a result my iPhone now has some physical damage.
Would the Apple store help with repairs/replacement if an iPhone experienced trouble due to an app?
Trouble due to an app would be software based so they would replace the software. Physical Damage would be damage. doesn't matter to them that you tossed your phone against a wall because of frustration with a stupid app.
Actually it might so I wouldn't mention that. intentional damage is grounds to deny service. So if you gave the tech a reason to be an ass they might use that against you.
As for getting coverage even with damage there are only three times they have any leeway, case by case, to cover something
1. very small hairline crack with no impact points that they truly feel was an accident and you aren't trying to scam them. and there are no other issues
2. slight and external only LCI tripped and totally unrelated to issue. Say like a speck of color in the headphone jack but your issue is that the battery is defective and there is zero sign of any liquid in the charging dock or battery connectors.
3. history recorded in their system of previous issue that mandates repair or swap and happened prior to damage. And no other damage exists. so say like i come in with an iPhone that is having a display issue. keeps going weird colors and the touch stops working in places. Needs a new display put on. But I'm paranoid about my data and I'm not sure I want them to open my phone unless i'm 100% positive I have a backup and I don't want to use iCloud. so i take it home to do it and come back in a day or two. Tech puts in a note so I don't have to waste time retesting, etc. But then my kid drops it and the screen gets all cracked up. Because of that note, they can overlook the damage and not charge me. If I hadn't been in, even if I said there was an issue before I dropped it, the drop takes precedence. Same if it went into water in the meantime because that causes damage to other parts.