Does any species, in any age, in any of the works by Tolkien have designated days of celebration, except birthdays, besides the holidays mentioned in the article about the Middle Earth Calendar on Wikipedia?
Appendix D to The Return of the King lists the holidays from the article you linked above, namely three Lithedays (roughly Midsummer) and three Yule days (Midwinter). It goes on to mention that during the reign of Elessar the calendar was reordered, to commemorate the fall of Sauron on March 25th, and September 22nd, Frodo's birthday, "was made a festival, and the leap-year was provided for by doubling this feast, called Comare, or Ring-day." (Return of the King pg 447)
There is no record of the Shire-folk commemorating either march 25 or September 22; but in the Westfarthing, especially in the country round Hobbiton Hill, there grew up a custom of making holiday and dancing in the Party Field, when weather permitted, on April 6. Some said that it was old Sam Gardner's birthday, some that it was the day on which the Golden Tree first flowered in 1420, and some that it was the Elves' New Year. In the Buckland the Horn of the Mark was blown at sundown every November 2 and bonfires and feasting followed." (Return of the King, pg 447-448)
It is worth noting that April 6 is listed in Appendix B of the same book as being when "The Ringbearers are honored in the the Field of Cormallen" and the next year as when "The mallorn flowers in the Party Field." November 2 is when Gandalf and the Hobbits "come to Bywater and rouse the Shire-folk." (Return of the King, pg 430-431)
On the island of Tol Eressëa, Elves celebrated a winter festival called Turuhalmë or "Log-Drawing". This festival is only mentioned in The Book of Lost Tales, so may not be considered canon.
Here are the relevant quotes:
Twill be a fitting day,' saith Lindo, 'for the sports of the morning in the snow and the gathering of the logs from the woods and the songs and drinking of Turuhalmë will leave us of right mood to listen to old tales beside this fire.' [...]
At length the day of Turuhalme was come, and the company from Mar Vanwa Tyaliéva went into the snowy woods to bring back firewood on sleighs. Never was the Tale-fire allowed to go out or to die into grey ash, but on the eve of Turuhalmë it sank always to a smaller blaze until Turuhalmë itself, when great logs were brought into the Room of the Tale-fire and being blessed by Lindo with ancient magic roared and flared anew upon the hearth.
~The Book of Lost Tales, History of Middle-earth volume 1