There are no rules that enforce sleep, but a mechanic of penalizing lack of sleep exists in the PHB.
There appear to be no rules in the PHB regarding just how much a character needs to sleep. However, it can be inferred from the description of the Trance feature that non-Elf creatures require up to 8 hours of sleep. Nothing specifically states what happens if the time spent sleeping is less than that, though. There is only a suggestion for skipping sleep entirely:
In the description of Constitution (PHB page 177) it says that the DM may request a character to roll a Constitution save if it tries to "go without sleep".
Whether the same thing could be done for sleeping a fraction of the full 8 hours is never mentioned, and probably left to the discretion of the DM.
Suggestions for house-ruling required sleep:
Seeing as Elves seem to be the only exception among the playable races of the standard edition, we may presume every other race behaves similarly to humans. And we know grown humans require between 7 and 9 hours of sleep (though this is a point of continued research and debate). The average of that is 8 hours, which neatly corresponds to the "8 hours for fully rested" principle inferred from the Trance feature.
One could then assume that a long rest of 8 hours of which a non-Elf character spends 2 hours standing watch and 6 hours sleeping will successfully complete the long rest, but leave the character a bit tired and wanting more sleep. One could also assume that not sleeping at all, while still technically a long rest, will leave the target exhausted. How to transfer this into rules and what to do with the states in-between is at the discretion of the DM. Usually the DM just lets you feel rested unless a special circumstance comes up. If you want sleep to matter in your campaign, though, I would suggest making a house rule. It could be something like:
If the character performed any strenuous activity during the day, he would require 8 hours of sleep to feel fully rested. Sleeping less than required leaves the character tired. If the character sleeps less than 8 hours a day for a number of consecutive days, the exhaustion accumulates. Once the character accumulated a total of 8 hours of loss of sleep, the character rolls a Constitution save. On a failed save, the character suffers one level of Exhaustion. Spending at least 8 hours sleeping or spending a day in light activity resets the counter.
This would mean that, if you were to spend only 6 hours sleeping every day during heavy adventuring, after the fourth day you'd have to start rolling Constitution saves (2 hours of sleep lost each day * 4 days = 8 hours of sleep lost). For a 7 hour sleep regime you'll be safe for 8 days. After you finally get your 8 hour sleep, you feel like you've caught up and that you're well rested. Another way to reset the sleep loss counter is simply traveling (without a forced march) and/or only engaging in other non-strenuous (primarily non-combat) activities during the day.
Elf's racial feat would then have an actual purpose. The same rule would apply, except for requiring only 4 hours of trance. Leaving the rest of the rule unchanged would then mean that an Elf would have to spend two consecutive nights not sleeping at all before he's forced to roll any Constitution saves (4 hours of sleep lost each day * 2 days = 8 hours sleep lost), and to reset the counter he'd just have to take a 4 hour snooze.
If counting missed hours of sleep seems like a tedious task, a simpler rule might suffice:
If a character slept less than their required amount (4 hours for Elves, 8 hours for the rest), they roll a Constitution save. On a failed save, the character suffers one level of Exhaustion. The DC for the save is reduced by 1 for each hour that the character spent sleeping.
As a DM you would have to decide what the base DC is. Here's a handy chart for a range of difficulties that you can use depending on the situation. For instance, the check should be easy at first, but gradually (point by point) get harder as the adventurers continue to miss on their sleep.