Would you be okay if a quest that could have any time based component at all have an icon showing that there isn't unlimited time for this quest? Or are you asking for a pop-up every time some game state is about to change?
Time limits have to be communicated in some way...and have to be consistent. If at the start of the quest you are told you have five days or something then enforce that, if the quest says you must do this now, no time to waste, fine with that. But what you cannot have are arbitrary invisible lines during quests that flip an invisible switch from infinite time, to no time without communicating that to the player in any way.
I actually wouldn't mind this concept. For example: you find a man in a cage that says he's starving or something and your character does "Oh no, he won't survive another night if we don't get him out of there" and the quest is marked "Urgent" or something meaning that if you long rest when you have an urgent quest, you fail it. But hey if you pass it and have no urgent quests at hand, sleep for a week.
I do think it would be a nice way for players to know they are on immediate, semi-immediate, and regular quest lines. maybe not so in your face about it, but just by looking at the active quests you have, it will show which quests are which.
