No, it really doesn't matter to the hotel, in general. It might take a few hours, but the hotel will eventually figure out the room is vacant and check out the guest. Even the most mediocre of hotels will have procedures to handle vacant but still checked-in rooms.

That said, it might matter to the next guest. A hotel can't clean a room until it's vacant. If you want to help out a stranger who wants to check in early, letting the hotel know you're leaving is an easy way to (possible) make someone else's day a tiny bit nicer.

If you can't check out on the app, just pick up the in-room phone and dial "0" for the operator. Tell whoever answers that you're checking out. If no one answers immediately, hang up after 5 or 6 rings knowing that you made a good-faith effort and just leave.

[The above advice is from someone who worked in hotels in various positions, including the Front Desk.]