[cookiecaper]

That's the trademark side of the case, and yes, In N Out could probably successfully argue that usage of their logo implies endorsement and/or is likely to create consumer confusion and that it's therefore a trademark violation. That's pretty easy for DoorDash to put to rest; just stop putting In N Out's logo on the site. A few good, highly-visible disclaimers would probably put any remainder of a trademark claim to rest.

A modified menu would be tougher. You can argue that there may be copyright infringement, and that may be valid to a limited extent, but DoorDash could put up a version of the menu that wasn't copyrighted, as its heart, menus are factual arrangements, and facts are not original creative works subject to copyright (though a specific arrangement of those facts may be). In N Out could also claim that there's trademark infringement if DoorDash is using trademarked food names, but I think it'd be a harder argument to win, since DoorDash can argue it's a nominative use.

IANAL.

[chrischen]

I can understand the factual part, but only if they didn't alter the prices. They'd have to at least mark it clearly that they're charging on top of the prices some amount.

[sharemywin]

they probably would need to strip the fluff from the menu. ie. the hot juicy blah burger with .... is just burger, ketchup, mayo, lettuce.