The scripts from Game of Thrones season 8 suggest that Jon Snow doesn't sleep. The HBO series officially came to an end after a six-episode eighth season this past spring, with showrunners, David Benioff and D.B. Weiss receiving a lot of criticism for the way they approached the fantasy series.
Jon's lack of sleep seems to be directly related to his connection with the Lord of Light. In the Game of Thrones season 5 episode, "Mother's Mercy," the Lord Commander of the Night's Watch was preparing a plan regarding the White Walkers. Ser Davos urged him to obtain the aid of the wildlings, much to the chagrin of the men of the Night's Watch. He was then lured outside, where members of the Watch stabbed Jon to death after declaring him a traitor. The following season, the Red Priestess Melisandre resurrected Jon using magic that she credited to the Lord of Light.
After coming back to life, Jon relinquished his duties at the Night's Watch and set out for taking back Winterfell before focusing on the bigger picture. Jon wasn't the only one in Game of Thrones to be resurrected by the Lord of Light, however. Beric Dondarrion was brought back to life six times by his friend Thoros of Myr with the use of the Lord of Light's magic. When Beric and the Hound were gearing up for the Battle of Winterfell in the episode, "A Knight of the Seven Kingdoms," Beric had an interesting line that was cut from the episode which was revealed by the scripts available at the Writers Guild Foundation Shavelson-Webb Library in Los Angeles (via Insider). The Hound urged Beric to get some sleep before he responded by saying: "Haven't slept since the first time I died. No point starting now."
Experiencing death and being resurrected was never a focal point in the series from the eyes of Jon. There was never any kind of explanation into how the events affected him physically or mentally. The ritual carried out by Melisandre gave him a new shot at life by healing his wounds but any other implications were left out of the series. Assuming that a resurrected human no longer needed to sleep as Beric suggested would create many logical questions. If this was the case, viewers would have surely wanted to know more about the magic's impact on Jon and the power stemming from the Lord of Light. People need sleep to survive so did that mean Jon wouldn't have needed food and water as well? How else might he have changed?
Some of the Game of Thrones season 8 script adjustments were understandably warranted but others seemed like a missed opportunity. There was more information regarding the relationship between Jaime and Brienne and their night spent together. The scripts also explained what Bran was doing during "The Long Night" which would have benefited the character's narrative. Most importantly, the scripts better explained Daenerys' feeling of isolation while in Winterfell and provided more insight into her mindset before demolishing King's Landing.