A possible problem I see with this, is that removing a players abilities might do more to make them feel weak during the wrong conditions, rather than feeling a bonus during the right conditions. However, it could be cool if it feels like a cycle that gives you rotating bonuses, so you always had some unit or spell with a good bonus. Or if the cycles went by frequently, so you just had to wait a turn or two for your spell to be available again.

*Update: You could probably focus on making the good alignments feeling like a power-up rather than a penalty when the alignments aren't correct. Make sure players have a good variety of abilities when the alignment isn't in their favor. Maybe you could have a more traditional magic system in place, like mana that runs out, forcing characters to carefully choose which abilities they use. But when the alignments are in your favor, mana replenishes freely, allowing that character to use their powerful abilities repeatedly. This could open some interesting strategic options, as you might want to focus down a mage whose magic is aligned. Maybe you could include spells that change the alignment on the fly, affecting all characters with the change.

removing a players abilities might do more to make them feel weak during the wrong conditions, rather than feeling a bonus during the right conditions. However, it could be cool if it feels like a cycle that gives you rotating bonuses, so you always had some unit or spell with a good bonus. Or if the cycles went by frequently, so you just had to wait a turn or two for your spell to be available again.

A cycle of many days means waiting many days for the right night to do things rather than going forward ASAP with one's plans. This fits well with many roleplaying game situations where the player either takes initiative or has the means to plan and respect an imposed schedule:

• You want to attack the evil mad wizard's fortress during the 4 hours window every 1440 days in which the defensive spells turn off. You have five weeks to get there from another country at an expected average travel speed of 16 miles per day. If you travel too slowly, or if you are delayed during the raid, the world is doomed.
• You want to seduce a fellow student at the wizards academy, where charm spells, countermeasures and counter-countermeasures are a standard part of courtship. Depending on your magical resources and what you have scouted of theirs, what's the optimal day to ask them out between now and the summer break?
• Today we have fire spells, let's torch giant spiders in their webs. On Wednesday we'll have enough ice magic to face the firebreathing dragon.

If the schedule is out of control, the game is going to be unfair.

• You discover you should be at the evil mad wizard's fortress tonight. No teleportation? Sorry, you've just lost the war.
• You want to seduce a fellow student at the wizards academy. Unfortunately you'd need several months of study to learn enough love magic for a good first impression. Sorry, wrong course plan at the start of the year.
• Today we have fire spells, the firebreathing dragon attacking us is laughing with a hint of sarcasm because he knows.

Hmm... An interesting idea, I do think.

What I'm inclined to suggest is that each moon (or combination of moons) be associated with a set of spells--and, importantly, that no one set of spells is inherently weak in some situation. Thus, instead of elemental affinities or bonuses, each set of spells would have different tactical properties. For example: the Black Moon is associated with stealth spells (including invisibility, instant-death spells, spells that silence or bind opponents, and so on); the White Moon is associated with the power of light (including blinding spells, burning rays, magic-missiles, and caster-focussed area-of-effect spells, etc.); and the Blue Moon is associated with spells of change (including spells that turn enemies into sheep, or that charm them, spells that turn the caster into a powerful melee creature, and spells that teleport characters, and so on). No enemies would be immune to these spells, so there's no "elemental rock-paper-scissors" to worry about, but each has different tactical properties, and thus calls for the player to adjust their play-style according to the state of the moons.

You want to attack the evil mad wizard's fortress during the 4 hours window every 1440 days in which the defensive spells turn off. You have five weeks to get there from another country at an expected average travel speed of 16 miles per day. If you travel too slowly, or if you are delayed during the raid, the world is doomed.

Something like that could be the premise for the whole game. Like the time limit in fallout 2, limiting how much extra traveling you can do before going to the last boss.

I see three ways this can be used:

-Long term planning, alignments affect game world (eg some magic door opens or something is weak/strong or some weird flowers blossom every 100 years), or your ability to do certain things. So you need a plan for when to be where and be prepared.

-Battle strategy dependent on time - assuming you have a variety of magic tools (not just single 'alignment'), you could easily have so many moons/planets that the configuration is never the same, and as time advances the battle mechanics change. Probably a balancing nightmare but at least it remains interesting. More advanced players can predict the alignment into the future to plan their strategies in advance, assuming its deterministic.

-Hiding the planetary mechanics from players to an extent, so finding items to measure/predict their positions could be an entire game on its own - you could take it as far as having to manually record data and do calculations (which would be great for creating a niche following for the game, but obviously should not be something the average player must do in order to play)