The Nitei Gavriel perek 79:7 writes, that there are those that throw the aravot onto the aron kodesh. On the next line, he writes:
וטוב להצניעם עד עת אפיית המצות כדי להשליכם בתוך האש שאופין מצות, הואיל
ואיתעב בה, חדה מצוה ליתעביד בה מצוה אחרת.
"And it's good to hide them until matza backing time, in order to throw them into the fire, being that one mitzva was done with them, it's good to do another one using them."
I understand the above in the following way. "וטוב - it's good", meaning that it's not a must, but a good thing to do.
The next word is "להצניעם - to hide them" meaning that even though they were thrown onto the Aron, they should now be put away in a different place until pesach.
As for the opinion of burning them with the cometz, The Mishnah Berurah (445.7) writes that if you have the Hoshanos, it's good to burn them on Erev Pesach with the chometz, being that one mitzvah was done with them, therefore one should also perform the mitzva of destroying the chometz using them.
אם יש לו הושענות טוב לשרוף החמץ בהושענות הואיל ואיתעביד בו מצוה חדא
ליתעביד בו גם מצות תשביתו
Is this a halacha or a minhag, or neither (e.g. "laziness"?) Does it need to stay there until that time
It's a minhag to burn them, but they don't have to stay up there until pesach, rather, you can put them away in a different place.
must it specifically be burned with the chametz or can one just
dispose of it at any time after Hoshanah Rabbah?
The nitei gavriel says in 79:6 that one should not place the aravot in a place where people might walk on it, or in the garage.
So it definitely has to be disposed of in an orderly fashion, and so burning it with chametz or baking matza with them seems like the best option.