Melisandre's necklace is no exception. Way back in the premiere of Season 6, we caught our first glimpse of the elderly version of the Red Woman when she removed the necklace and readied herself for bed, revealing that she was actually a much older woman than she appeared. Her exact age isn't known, but it's safe to assume that she's seen a few centuries pass. The implication of that scene was that the necklace functioned like a youth-giving chalice and provided her with strength, so it makes sense that abandoning it, like she does at the end of "The Long Night," would have disastrous effects on her health.
But, as many fans online pointed out soon after the reveal scene in Season 6, there was a moment in Episode 7 of Season 4 when Melisandre was shown without her necklace taking a bath. She didn't suddenly turn into the older version of herself. That's an understandable mistake: Errors happen, and it's certainly possible the necklace wasn't always identified by the writers as a type of youth-preserving shield. In an interview with Elle back in 2017, actress Carice van Houten identified it as "an oops moment."
Of course, for some corners of the internet, that explanation wasn't enough, and more elaborate explanations were invented. "The necklace could just be a catalyst/enhancer for her magic," wrote one fan on Reddit last year. "She lost her faith after Stannis lost and she couldn't keep the appearance without the necklace. Before that her faith was strong enough to keep the 'simple' appearance magic up without [the] necklace." As with most Game of Thrones theories, there's no shortage of speculation if you go looking for it.
With Melisandre now dead, it's unlikely we'll be treated to a flashback explaining the origins of the necklace in the last three episodes; there are at least a couple more pressing plot threads to resolve before the series comes to a close. Similarly, the light going out on the necklace suggests that it probably didn't retain its magic and was somehow tied to her specifically, so don't expect some other character to randomly pick it up and suddenly acquire her powers of sorcery and permanent youthfulness. It feels more likely that the necklace will die with her, taking its mysteries to the grave.