I usually enjoy environmental effects in D&D. It adds variety to each game session, and it keeps the players on their toes. For instance, when I run games in Barovia, I have players roll a d6 every new day (as long as Strahd is in power.) They get a penalty to the corresponding attribute during that game day. It makes the environment oppressive, it changes up how well their character performs from game session to game session, it makes the monsters more or less powerful, and it gives urgency to the mission. Depose Strahd if you ever want to play your character at full power again.
In AD&D 2nd Edition Dragonlance, there's a similar mechanic for magic.
5th Edition Dragonlance makes an attempt to suggest this mechanic through the Lunar Sorcery subclass.
I sat down with my AD&D Dragonlance Adventures source book and the 5th Edition Shadow of the Dragon Queen and compared the two systems.
In 5th Edition, there's no superclass for magic users, so awkwardly the three moons of Krynn only effects Sorcerers. If you're a wizard or warlock, the moons have no effect on your magic. And even for sorcerers, this is an option. It's a subclass. You don't have to take it, so possibly in some campaigns the moons of Krynn have no effect on anything. I feel like that's a big loss to worldbuilding, because the constellations and the moons are a really big part of Krynn's flavour. The gods literally dwell up in the stars, a little like Nyx in Theros. If you look up into the sky and see a constellation missing, it's because that god is down on Krynn walking around somewhere. The moons are white, red, and black, not coincidentally the colour of the robes of the Wizards of High Sorcery (that's the Mages of High Sorcery in 5th Edition.)
The Lunar Sorcery subclass functions as a hot-swappable group of spells. It's a pretty cool mechanic. When there's a full moon, you get one set of spells. When there's a crescent moon, you get a different set of spells. As a group, depending on your level. So a sorcerer can swap out spells with this subclass, and that seems like a lot of fun to me.
Unfortunately, it's up to the player to decide the phase of the moon on any given day. That seems odd to me, because it can mean that a moon just never changes for a whole campaign. Or a moon could go from full to new to crescent within a matter of just 3 days.
It's also strange that there's no distinction of which moon you're drawing power from. All of the moons provide the same benefit, I guess.
I like the idea of granting groups of spells based on some in-game environmental effect, but what I don't like:
I honestly can't see myself using these rules. I'd rather re-implement the AD&D; rules, or at least I'd track the phases of the moon manually, because the 5e mechanic doesn't even fulfill its own promise. This isn't lunar sorcery, because the player gets to control the moon. This is just really just giving spell scrolls to a sorcerer as a package deal, and calling it lunar magic.
Luckily, AD&D has a fix for this.
Wizards of White Robes of 3rd level or above get their power from the moon Solinari.
Wizards of Red Robes of 3rd level or above get their power from Lunitari.
Wizards of Black Robes of 3rd level or above get their power from Nuitari.
If "your" moon is in a specific phase, there are some specific effects:
1 additional spell
When two moons are aligned (you can trace a straight line through one and then through the other), there are additional effects.
There was a chart in the book that helped you track the position of the moons. Each game day, you moved the moons and applied any effects.
It sounds like a lot of work, but in practice it's just part of good timekeeping, which isn't hard, but probably deserves a post of its own eventually.
You can download my updated lunar tracking chart from DMsGuild.com.