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.
Generally speaking prisoners are provided some nourishment or they'd be killed rather than kept as prisoners. On average a human being can live around 3 weeks without food or around 3 days without water. So you actually could take a long rest before freeing the prisoners and they'd be fine...a long rest is considered around 8 hours in D&D which is about a 3rd of a day or around a 9th of the time it would take someone to die of dehydration.
I'm just saying if something is time sensitive, then show that on the quest list as urgent or immediate.