Are you talking about the books or TV show?
What is Davos fate?
Books, can't predict the show. You've asked about one of my blind spots, his ultimate fate I'm not sure.
When Jon declares KITN Davos will be his Hand. [1] When Jon marches south of Tyrion's throne a climactic battle will occur at the God's Eye, there Davos will be part of an archery team (lead by Arya) that will bring down Viserion ridden by Tyrion, probably with weirwood arrows from the Isle of Faces. Viserion will die [2].
The narrow sea will literally run dry [3] and Drogon will turn to stone somewhere around Dragonstone. Dany will be searching for Drogon to wake him from stone. And Jon will go searching for Dany to prevent her from doing it. Davos is going to locate Stone Drogon for either one of Dany or Jon, I'm not sure but I think Jon, I think Dany will find it herself and Davos locates Drogon and Dany there with the dragon in time. And Davos will fight with Jon against Dany's sworn swords to prevent her from waking Drogon from stone. At some point in the dried out narrow sea Davos is going to find his fingerbones, his luck, I'm thinking before he locates Stone Drogon. [4]
After that I don't know, maybe hand for the IT at the end (Jon won't be king).
[1] Besides it being apparent Davos is a good Hand type of guy, Daeron the Young Dragon foreshadows Jon. Jon will fall into the KITN role when Stannis falls. Jon will keep Davos on as hand as Daeron kept on his predecessor's hand.
[2]
In those centuries of trial and tumult, the Reach produced many a fearless warrior. From that day to this, the singers have celebrated the deeds of knights like Serwyn of the Mirror Shield, Davos the Dragonslayer, Roland of the Horn, and the Knight Without Armorand the legendary kings who led them, among them Garth V (Hammer of the Dornish), Gwayne I (the Gallant), Gyles I (the Woe), Gareth II (the Grim), Garth VI (the Morningstar), and Gordan I (Grey-Eyes).
Aemond Targaryen is a Tyrion parallel by virtue of kinslaying, mismatched eyes, said to be half the size of his brother on his birth, etc. Aemond died by way of Dark Sister through the eye, above the God's Eye. He fell in. Dark Sister for Arya.
The dragon Tessarion (name association with Viserion) was put down with an arrow through the eye.
Black Aly is an Arya parallel. She lead the lads, particularly a contingent of archers, out of the Riverlands during the end of the Dance when Cregan Stark and the Winter Wolves descended on the throne, successfully fighting their way through the Riverlands and at the God's Eye. Think the BWB remnants, Arya's fantasy that she could return to them like Wenda the White Fawn (a the leader of the previous Riverlands rebellion) that Anguy could teach her archery.
Tyrion to Sansa. Foreshadowing for a northman bringing down his dragon in the Riverlands. He'll survive by falling off into the God's Eye."One of your northmen hit me with a Morningstar during the battle on the Green Fork. I escaped him by falling off my horse."
Davos' sigil is a ship and sails."Well, how long does a dragon live?" She looked up as Viserion swooped low over the ship, his wings beating slowly and stirring the limp sails.
General foreboding great eye for Tyrion.A full moon floated above the mast. It is following me downriver, watching me like some great eye. Despite the warmth of the musty skins that covered him, a shiver went through the little man.
[3]
And all the narrow sea once being a much smaller inland lake speculation in TWOIAF."When the sun rises in the west and sets in the east," said Mirri Maz Duur. "When the seas go dry and mountains blow in the wind like leaves. When your womb quickens again, and you bear a living child. Then he will return, and not before."
[4] Dany waking the dragon and stone Drogon is its own issue.
Relating specifically to Davos, Davos is foreshadowed by Tom Tangletongue. The bloke who wasn't comfortable speaking due to his stammer as Davos isn't comfortable speaking in high company because of his low birth. Tom thus listened more than he spoke and was wiser for it, like Davos. At the end of the dance Tom worked out Aegon's dragon was still alive by plying sailors with alcohol and listening to them, the same method Davos uses in the Belly of the Whale. Tom locates the dragon for his king.
Drogon will turn to stone and also become a Sea Dragon. These two are heavily associated with Davos at Dragonstone, by way of Sea Dragon Tower where he learns to read and the stone dragons. This is the choicest quote.
Generally the plot will retread Stannis and Mel trying to wake the dragon with Edric, but with Dany playing their part and Davos replaying his, helping Jon to prevent it.Davos had often heard it said that the wizards of Valyria did not cut and chisel as common masons did, but worked stone with fire and magic as a potter might work clay. But now he wondered. What if they were real dragons, somehow turned to stone?
Jon stopping Dany I think will parallel the TOJ battle, it will parallel Aegon taking Dragonstone from Rhaenyra.The fingerbones is just connecting the sea running dry to where Davos lost them and where he'll be in future, and that he keeps harking on having lost his luck. I don't know where or when but Davos will save Arya at some stage, as Tom saved Arya parallel Lady Baela.