He was "lost" and never returned to Doriath
The only mention of Daeron in The Silmarillion after the two betrayals mentioned in the question is:
Upon Doriath evil days had fallen. Grief and silence
had come upon all its people when Lúthien was lost. Long
they had sought for her in vain. And it is told that in that time
Daeron the minstrel of Thingol strayed from the land, and
was seen no more.
The Silmarillion: Quenta Silmarillion, Chapter 19: Of Beren and Lúthien
Page 183 (George Allen and Unwin 1977 hardback edition)
There are other versions of the story that mention him. One of Tolkien's earliest writings is The Tale of Tinúviel, in which Daeron's place is taken by Tinúviel's brother, Dairon. It is he who betrays Beren and Tinúviel. In this version, he also becomes lost.
'Tis said that Dairon sped swiftest and furthest in pursuit, but was wrapped in the deceit of those far places, and became utterly lost...
The History of Middle-earth Volume II, Chapter I: The Tale of Tinúviel
Page 46 (Houghton Mifflin 1984 hardback edition)
Dairon reappears in Tolkien's later writings about the First Age. In The Grey Annals, he again betrays Lúthien and is "lost". When Beren and Lúthien return to Menegroth with the Silmaril, we are told:
Glad was their welcome in Doriath, for a spell of shadow and silence had lain upon all since Lúthien fled; and Dairon seeking her in sorrow had wandered far away and was lost.
The History of Middle-earth Volume XI, Part One: The Grey Annals
Page 69 (Harper Collins 2015 paperback edition)
The common factor in all these versions of the story of Daeron/Dairon is that he is sorry for his betrayal, or at least that his betrayal led to Lúthien's departure. He leaves in search of her and is never seen again.