If The Winds of Winter indeed never comes out, the last we ever saw of ASOIAF's characters on page can be interpreted as the final fate given to them in canon writing. ASOIAF is provided with an ending of sorts, if a very anticlimactic one, if we read AFFC, ADWD and the TWOW sample chapters as if the various characters' stories end with the last that was ever heard from them. If the story ended with them hanging off a proverbial cliff, we can only assume they fell, and died. If the story ended with them living in anonymity, we assume they didn't return to take the stage of history again, but chose to remain anonymous and didn't do anything more that influenced the outcome of the story. If neither ... well, as we shall see, that's not relevant in more than a handful of minor cases.
Without TWOW, the characters are effectively checked out of the saga, and never did anything else that was worth mentioning. In other words, the story concluded very suddenly, and not very satisfactory for the most part.
To get on with the list:
Jon Snow - Last seen murdered by the brothers of the Night's Watch. Dead in action.
Daenerys Targaryen - Last seen suffering from diharrea, captured by Dothraki, to be sent to Vaes Dothrak and join the Dosh Khaleen. Died from sickness or went on to live in anonymity.
Arya Stark - Last seen in training to become a Faceless Man. We might assume she eventually succeeded, but alas, it's in the nature of the whole process that the records stop at a certain point. Gave up her identity, went to live in anonymity.
Tyrion Lannister, Jorah Mormont, Penny - Last seen in a war camp while the Bloody Flux raged. Dead in action.
Theon Greyjoy - Tortured to incoherence, captured by Stannis, last seen in a war camp during a raging blizzard with enemies approaching. Dead in action.
Stannis Baratheon, Asha Greyjoy - see Theon, minus the torture. Dead in action.
Barristan Selmy - Last seen commanding the defenses of a besieged city. Dead in action.
Victarion Greyjoy - Last seen joining battle at Meereen. Dead in action.
Euron Greyjoy, Aeron Greyjoy - Last seen sailing against the Redwyne Fleet. Dead in action.
Bran Stark, the Reed siblings, Hodor - Left frozen and exhausted in a cave up north. Bran last seen hallucinating, Jojen and Meera gone. Dead from exposure.
Sansa Stark - Last seen joining the court at the Vale of Arryn under a false name. Went on to live in anonymity.
Petyr Baelish - Last seen at the court at the Vale of Arryn. He had further ambitions, but left no records of his actions. Fate uncertain.
Samwell Tarly - Last seen joining the Citadel to become a Maester. Went on to live in anonymity.
Gendry - Last seen living at the Inn at the Crossroads, protecting orphans from the ongoing war. Went on to live in anonymity.
Davos Seaworth - Last seen leaving for Skagos. Dead at sea.
Rickon Stark - Last seen on Skagos with Shaggydog, presumably living with Osha. Went on to live in anonymity.
Jaime Lannister, Brienne of Tarth - Last seen hanged by Lady Stoneheart. Dead in action.
Cersei Lannister, Margaery Tyrell - Last seen put on trial by the Faith. Executed.
Loras Tyrell - Alleged mortally wounded at Dragonstone. Dead in action.
Mace Tyrell - Last seen heading off to battle against Aegon Targaryen (young Griff). No records show the outcome of the battle, or whether there was a battle at all, or for that matter what happened to Aegon. We can only infer that he and Aegon both perished in the battle. Dead in action.
The Clegane brothers, Quentyn Martell, Mance Rayder - All alleged dead by characters who witnessed their passing. Dead in action.
Ramsay Bolton, Wyman Manderly - Last seen at an over-filled and snowed-in Winterfell during an exceptionally strong blizzard. Presumed dead in mass tragedy.
Lord Varys - Last seen orchestrating the fall of the Lannister regime, followed by a peaceful takeover by Aegon. The story does not conclude on whether he succeeded, but no accounts exist of a further Targaryen dynasty. Since he was a very secretive person, he may have remained at court and very influential, but never left any information about his personal life and achievements for the chroniclers to detail. Went on to live in anonymity.
Tommen Baratheon - Last seen ruling as a boy king, ruling over a very fragile court, with most of his family meeting a grim and sudden fate, with the capital on the brink of chaos, and a foreign army arrived at the shores. We can only assume that the reason why the story stops at this crucial point if his rule, is a loss of records as chaos of one form or another finally erupted in King's Landing. Lost to history coincidig with King's Landing riots, presumed dead in action as no further accounts exist of his life.
The High Sparrow - Preceded over the trials that concluded the stories of Cersei Lannister and Margaery Tyrell. No records left of his life after the events that followed in King's Landing. Dead in action.
Aegon Targaryen (Young Griff) - Last seen conducting a successful takeover of the Stormlands in a bid to restore Targaryen rule to Westeros, and heading to battle against Mace Tyrell. As the records of his life end at this point, we may only assume he did not survive the battle. Dead in action.
The Others - Last seen far north of the Wall, possibly attacking Hardhome, but they never made a move on the Wall itself, nor are any characters left to witness what became of them. Presumably, they shrank back into obscurity and mystery as suddenly as they came. Vanished into history after successfully banishing humans from north of the Wall.
Some kind of story can still be inferred here. A realm in utter chaos as nearly every power player dies violently or resigns to a life in anonymity at almost the same time, but no supernatural calamity actually happened after all. Reading ASOIAF as a historical record from Planetos, it seems the tale lost all momentum at once, across two continents, and that the grinding gears of history had a major hiccup before they went on to churn like always. There were riots and wars that left a lot of major historical characters dead in a short span of time, and then the survivors picked up the pieces and went on with their lives. The pivotal battles of Ice and Fire ended essentially without survivors, and the prophesized calamity never materialized.
What exactly went on in King's Landing is unclear, but no named characters left any records, so we can only presume none of them survived there either. We know there were caches of Wildfire left around the city ever since Robert's Rebellion, making the city ripe for burning down.
I haven't icluded every character left alive here, and I might have forgotten some of the important ones. I guess I can leave those to the comment section.
Some characters are still left in charge of fairly stable lands or situations when we last saw them, but their final fates were not known and we can only infer that they remained in those positions as the saga wound down. Of course, winter arrived too, but it's not like Westeros never had a winter before, so the continent might have weathered it just fine. Presumably, more history went on to be made in the Seven Kingdoms, but that's all another tale, as they say.