If I'm a level 5 druid (Circle of Dreams, from Xanathar's Guide to Everything). Am I limited to using two fey energy dice for my Balm of the Summer Court feature before having to take a long rest to use it again?
You're limited to two dice per use, but five dice per long rest
There's rules in the feature about only using half your dice, but not in the context of a long rest:
At 2nd level, you become imbued with the blessings of the Summer Court. You are a font of energy that offers respite from injuries. You have a pool of fey energy represented by a number of d6s equal to your druid level.
As a bonus action, you can choose one creature you can see within 120 feet of you and spend a number of those dice equal to half your druid level or less. [...]
You regain all expended dice when you finish a long rest.
—Balm of the Summer Court, Xanathar's Guide to Everything, pg. 22
So at level 5, the druid would be permitted to spend 2.5 dice per use, which we round down to 2 because of the general rule about rounding:
There's one more general rule you need to know at the outset. Whenever you divide a number in the game, round down if you end up with a fraction, even if the fraction is one-half or greater.
—Round Down, Player's Handbook, pg. 7
So between Long Rests, a Druid could use this feature five times, spending 1 die each time, or three times with 2 dice for the first two and 1 for the last, or any other permutation they choose that obeys these restrictions.
You currently have a pool of 5 d6s that you can use to heal people. Once you use all 5 you will need to take a long rest to use them again.
However, as a bonus action, you can only use a maximum of 2 of these dice. But that will still leave you with at least 3 to use before you need to take a long rest.