A previous revision of this answer suggested that long rests did not require sleep and there were no rules for sleep deprivation. These assertions were respectively rendered incorrect by the publishing of 2017's PHB errata and then Xanathar's Guide to Everything.
The core books (PHB, DMG, and MM) have no rules regarding penalties for sleep deprivation; the closest I can find is that the PHB entry for Constitution states:
The DM might call for a Constitution check when you try to accomplish tasks like the following:
However, Xanathar's Guide to Everything contains a wealth of optional rules for DMs who want to cover more unusual situations, and on page 78 it offers the following rule regarding characters who are forgoing long rests/sleep:
Whenever you end a 24-hour period without finishing a long rest, you must succeed on a DC 10 Constitution saving throw or suffer one level of exhaustion.
In any event; the application of levels of exhaustion to characters who are going without sleep is the obvious way to represent such a situation in the rules of the game. Your GM might rule that exhaustion incurred this way cannot be removed without sleeping; for starvation and dehydration, for instance, the game states that exhaustion caused by a lack of food or water cannot be removed until the character has eaten and drunk the appropriate amount.
In 3e, a popular method of overcoming fatigue was the use of the Restoration spells in their capacity to remove the conditions of exhaustion or fatigue. In 5e, the Greater Restoration spell can be used to remove one level of exhaustion (the Lesser Restoration spell only affects a much more limited set of conditions and cannot be used to combat exhaustion). Were I your GM and I was giving you levels of exhaustion for not sleeping, I would rule that Greater Restoration could be used to overcome sleep-related fatigue.
You might also have luck with the possibility of a custom magic item. 5e includes an Ioun Stone of Sustenance (and previous versions of the game included Rings of Sustenance) which enable the character to go without needing to eat or drink at all. A similarly priced item of Sleeplessness/Wakefulness which likewise negates the need for characters to actually sleep would seem like a perfectly reasonable magical item to me.