This history of the Dragon-helm is a bit complicated and, in some cases, incomplete. What happened to it after the Fall of Nargothrond is unknown.
The Helm was made by Telchar, a dwarf from Nogrod, who also made Narsil (later Andúril, Aragorn's sword) and Angrist (the knife used to cut a Silmaril from Morgoth's crown). The city of Nogrod would have been in the area of the Shire, but it was destroyed at the end of the First Age, during the War of Wrath. The exact date the Helm was wrought is unclear, but it was made soon after Glaurung appeared in 260 FA, about 200 years before Túrin was born.
It was made for the dwarf Azaghâl, Lord of Belegost, another city not too far from Nogrod. After Maedhros saved his life during an ambush by the Orcs, Azaghâl passed the Helm to him. Azaghâl was later killed by Glaurung during the Nírnaeth Arnoediad.
Maedhros then sent the Helm as a gift to Fingon, who had beaten Glaurung back to Angband. Finally, when Fingon gave the lordship of Dor-lómin to Hador, he also gave him the Helm and it became a heirloom of the House of Hador.
Húrin, father of Túrin, did not use the Helm in battle ("I would rather look on my foes with my true face"), but still kept it as a treasured item. When Thingol sent messengers to Morwen asking her to go to Doriath to be reunited with her son, she refused and sent back the Helm, but Túrin was not yet old enough to wear it.
At seventeen, Túrin took the Helm and wore it to battle on the northern borders of Doriath, in the region of Dimbar. After three years of fighting, he left the Helm in Dimbar and came back to Menegroth where he killed Saeros and fled. When Beleg decided to go back to Túrin on Amon Rûdh, he went to Dimbar first to retrieve the Helm and gave it to him.
Travel to Nargothrond
Túrin eventually ends up in Nargothrond with the Helm, but it is not clear how. Christopher Tolkien says this:
It may be noted here that it was my father's intention to extend the history of the Dragon-helm of Dor-lómin into the period of Túrin's sojourn in Nargothrond and even beyond; but this was never incorporated into the narratives.
In the existing versions the Helm disappears with the end of Dor-Cúarthol, in the destruction of the outlaws' stronghold on Amon Rûdh; but in some way it was to reappear in Túrin's possession at Nargothrond. It could only have come there if it had been taken by the Orcs that carried Túrin off to Angband; but its recovery from them at the time of Túrin's rescue by Beleg and Gwindor would have required some development of the narrative at that point.
Unfinished Tales, Narn I Hîn Húrin, p. 199
He would also say this:
My father must have supposed therefore that Gwindor and Turin carried it with them to Nargothrond. This raises the obvious difficulty that the Helm would at once have revealed the identity of Turin; but in Unfinished Tales (pp. 154-5) I have referred to an isolated piece of writing among the Narn material which 'tells that in Nargothrond Turin would not wear the Helm again "lest it reveal him", but that he wore it when he went to the Battle of
Tumhalad.' The passage in question reads:
Beleg searching the orc-camp [in Taur-nu-Fuin] finds the dragon-helm - or was it set on Turin's head in mockery by the Orcs that tormented him? Thus it was borne away to Nargothrond; but Turin would not wear it again, lest it reveal him, until the Battle of [Talath] Dirnen [and the destruction of Nargothrond].
The History of Middle-earth Volume 11: The War of the Jewels, The Grey Annals, Commentary, §275, p. 140
Then we get to the confusing parts. In The Silmarillion, the Helm is not in Nargothrond and Túrin uses a different "dwarf-mask":
Therefore they gave him dwarf-mail, to guard him; and in a grim mood he found also in the armouries a dwarf-mask all gilded, and he put it on before battle, and his enemies fled before his face. [...]
Then the warriors of Nargothrond went forth, and tall and terrible on that day looked Túrin, and the heart of the host was upheld, as he rode on the right hand of Orodreth. But greater far was the host of Morgoth than any scouts had told, and none but Túrin defended by his dwarf-mask could withstand the approach of Glaurung; and the Elves were driven back and pressed by the Orcs into the field of Tumhalad, between Ginglith and Narog, and there they were penned.
The Silmarillion, Of Túrin Turambar, p. 250, 253
Christopher Tolkien admits to having included this text in error:
In the published Silmarillion (p. 210) I adopted a passage from another text in the vast assemblage of the Narn papers, telling how Túrin found in the armouries of Nargothrond 'a dwarf-mask all gilded', and wore it into battle.
It seems probable that this story arose at a stage when my father was treating the Dragon-helm as lost and out of the story (from the end of Dor-Cuarthol, the Land of Bow and Helm, when Turin was taken by the Orcs), and I extended Turin's wearing of it to the battle of Tumhalad (p. 212).
The History of Middle-earth Volume 11: The War of the Jewels, The Grey Annals, Commentary, §280, p. 144
Other texts mention that Túrin did wear the Helm in the battle of Tumhalad:
Then Orodreth and Túrin and Gwindor went up against [Glaurung], but they were defeated upon the field of Tum-halad; and Orodreth was slain and Gwindor. [Pencilled in margin: Túrin in the battle wore the Dragon-helm.] Turin bore Gwindor out of the rout, and escaping to a wood there laid him on the grass.
The History of Middle-earth Volume 11: The War of the Jewels, The Grey Annals, §275, p. 85
It is true that up to this point, Túrin had tried to conceal his identity from the general public. He may have felt that the benefits of wearing the Helm in such a battle outweighed the consequences.
Meeting with Glaurung
There is also some confusion about the meeting with Glaurung near the doors of Nargothrond and whether the Helm could protect against the dragon's eyes. Initially, Túrin looked at Glaurung and "fell under the dreadful spell of the dragon, and was as one turned to stone."
But then a note was pencilled next to this:
For while he wore the Dragon-helm of Galion he was proof against the glance of Glaurung. Then the Worm perceiving this
The History of Middle-earth Volume 11: The War of the Jewels, The Grey Annals, §280, p. 86
The note abruptly ends there. Christopher Tolkien says this:
The further pencilled note here on the subject of the Dragon-helm, observing that while Turin wore it he was proof against Glaurung's eyes, can be somewhat amplified. [...] Here it is said that Glaurung desired to rid Turin of the aid and protection of the Dragon-helm, and taunted him, saying that he had not the courage to look him in the face.
And indeed so great was the terror of the Dragon that Turin dared not look straight upon his eye, but had kept the visor of his helmet down, shielding his face, and in his parley had looked no higher than Glaurung's feet. But being thus taunted, in pride and rashness he thrust up the visor and looked Glaurung in the eye.
The History of Middle-earth Volume 11: The War of the Jewels, The Grey Annals, Commentary, §280, p. 143
In any case, the story of the Dragon-helm ends here. There is yet another scrap of paper about Húrin's travels after being released:
Hurin goes to seek Gondolin. Fails. Passes by Brethil, and his anguish is increased. They will not admit him - saying that the Halethrim do not wish any more to become enmeshed in the shadow of his kin. But ??? [?new] Lord gives the dragon-helm to Hurin. His heart is hot against Thingol.
The History of Middle-earth Volume 11: The War of the Jewels, The Wanderings of Húrin, p. 255
The "???" above refer to a blank space which was for the name of the Lord of Brethil, but it was never filled in. How the Helm would have gone from Nargothrond to Brethil is unknown.
Finally, Tolkien did write a longish text about the wanderings of Húrin, but it does not mention the Helm at all.