Are Hallows and Horcruxes connected to Celtic mythology?
Author J.K. Rowling has openly stated that parts of the Harry Potter books are based upon Celtic and British mythology, as well as Greek myth and the Bible. Hallowed items are very common parts of the folklore and mythology of the British Isles, with variances on the tale from place to place. The Irish-Celtic tradition tells a story of an ancient race of people who brought Four Treasures from the Otherworld to the island of Ireland. Among these Four Treasures are two items which have counterparts in the Deathly Hallows myth—the stone and the wand. The Stone of Destiny is supposed to "cry out" whenever the true King of Ireland is nearby, and the Spear of Lugh is an unbeatable weapon. The Resurrection Stone, of the Harry Potter series, does not have the same abilities as the Stone of Destiny, but it is quite probable that Rowling liked the idea of a magical stone and incorporated it into her work. The Elder Wand is plainly drawn from the Spear of Lugh, and variations on the myth describe the spear alternately as a staff or wand.
The third Hallow, the Cloak of Invisibility, is drawn from a later mainland British mythology—the Arthurian legends. In this tradition, there are Thirteen Treasures of Britiain, including a mantle (also called a hood or cloak) which gave Arthur the power to be invisible.
As for the Horcruxes, the idea of using a physical object to store a part of one's soul for safekeeping in the event of bodily death is most likely drawn from Slavic mythology—as opposed to British or Celtic—and the tale of Koscei the Deathless. Koscei is a character of Slavic mythology who is known for hiding his soul in various animal or object vessels. In one telling, he hides his soul in a needle inside of an egg inside of a duck inside of a rabbit inside of a locked chest! Only by destroying this vessel can Koscei really be killed. This myth might've made its way to the British Isles in one of the many waves of Northern European migration, or Rowling might've come across it in her own studies.
The items Voldemort chose for his Horcruxes and their respective powers or special qualities may also be drawn from myth. Let's start with one which is reminiscent of the Koscei narrative—Nagini. Voldemort hid a portion of his soul in his magical pet snake, who could fight off or run from attackers. Again, this is a parallel to Slavic mythology, but snakes are such an important creature to Voldemort and the Harry Potter series that there's no better choice for an animal vessel.
As for the other six Horcruxes, several are highly reminiscent of British mythology. The Sword of Godric Gryffindor is special because it can only be summoned by a true Gryffindor and someone who is loyal to Albus Dumbledore. Swords are numerous in British mythology, and Gryffindor's sword has some parallels to the mythic Exaclibur or Caledfwlch. Only the true King of Britian could (variably) use, lift, or summon this sword.
Ravenclaw's Diadem, Slytherin's Locket, and Marvolo Gaunt's Ring do not have any precise parallels in British mythology, but these sorts of valuable items with special powers are a theme in many myths. Similarly, Hufflepuff's Cup does not have a precise parallel, but it does resemble many a mythic cauldron of British myth. Food and its multiplication are major themes in British myths, with several stories telling of magical cauldrons that multiply and/ or improve the flavor of any food put inside it. Hufflepuff's Cup does not multiply what is inside, but the copies of the cup from Gringotts Wizarding Bank's Vault multiply themselves when touched.
The diary and Harry do not have any parallels I am aware of in British mythology.
Just for fun, let's consider one more part of the Harry Potter series drawn from British mythology—wizard's chess! In the Harry Potter universe, witches and wizards enjoy passing the time by playing chess with figures who move themselves across the board as commanded. In British myth, Gwenddolyn (also spelled Gwenddolau) was a king and friend of Merlin who owned a golden chessboard with silver pieces who would move about it as instructed!