Yes, it appears that the stars moved
In the sense that they changed position in the sky during the course of a night and of a year. It's unclear whether, over longer timescales, the constellations would change and drift, as happens in the real world.
From Fellowship of the Ring, Chapter 3:
Away high in the East swung Remmirath, the Netted Stars, and slowly above the mists red Borgil rose, glowing like a jewel of fire. Then by some shift of airs all the mist was drawn away like a veil, and there leaned up, as he climbed over the rim of the world, the Swordsman of the Sky, Menelvagor with his shining belt.
Emphasis mine; Remmirath is probably the Pleiades, Borgil is likely Betelgeuse or Aldebaran, and Menelvagor is almost certainly Orion. Later...
From Lord of the Rings, Chapter 10:
Peering out, Frodo saw that the night was still clear. The Sickle was swinging bright above the shoulders of Bree-hill.
Notice that all of these descriptions evoke movement - swinging and climbing, just as the stars appear to spin in the sky during a night. Only one star is explicitly stated to be rising, but the similar language in the descriptions of the others suggests they were also climbing.
The significance of the crown of stars in Mirrormere was not that Durin looked in and saw a fluke alignment of stars that happened to look like a crown; it was that he looked during the day and saw the crown of stars. Moreover, everyone who looked thereafter saw the same sight - the crown of stars. From Lord of the Rings, Chapter 6:
They stooped over the dark water. At first they could see nothing. Then slowly they saw the forms of the encircling mountains mirrored in a profound blue, and the peaks were like plumes of white flame above them; beyond there was a space of sky. There like jewels sunk in the deep shone glinting stars, though sunlight was in the sky above. Of their own stooping forms no shadow could be seen.
There is an ancient belief that if one stands at the bottom of a deep well/hole/canyon/what have you, the stars are visible even during the day. Very likely this is what Tolkien was referring to with Mirrormere.
The importance of the site, then, was not that Durin looked and saw the stars, it was that the stars remained there. The dwarves, especially those of Durin's line, saw this as a promise that their kingship would endure.