Of the more important characters who die in the series, in particular in Deathly Hallows, do some of them have a ghost? I think they qualify as having unfinished business.
Ghosts in Harry Potter aren't the product of 'Unfinished Business' as they often are in other fictional universes; they are produced by someone dying in a state where they are terrified of death or unwilling to 'move on' and face what they might find in the afterlife. (There are some hard to explain exceptions, like Professor Binns, who we can make guesses about, but nothing canon is provided to allow us to be sure.)
Here's what Nick had to say in OoTP:
“Wizards can leave an imprint of themselves upon the earth, to walk palely where their living selves once trod,” said Nick miserably. “But very few wizards choose that path.”
“I was afraid of death,” said Nick softly. “I chose to remain behind. I sometimes wonder whether I oughtn’t to have… well, that is neither here nor there… in fact, I am neither here nor there…” He gave a small sad chuckle. “I know nothing of the secrets of death, Harry, for I chose my feeble imitation of life instead. I believe learned wizards study the matter in the Department of Mysteries—”
Snape also provided commentary on ghosts:
A ghost, as I trust that you are all aware by now, is the imprint of a departed soul left upon the earth... and of course, as Potter so wisely tells us, transparent.
The Harry Potter wiki has this comment:
As ghosts are imprints of souls of the said deceased wizards and witches, they are unable to move beyond the veil, and are forever trapped in the world of living as intangible beings.
It's quite possible that one of the characters left a ghost... but it's suggested that the characters we cared about probably didn't. The students who died in the last fight died 'with their boots on' as they say, fighting for something they believed in; it seems unlikely that those willingly risking their lives would leave behind ghosts -- they were committed to their fates and seem unlikely to spend their last moments trying to escape them.
As to Sirus, the same applies; for confirmation,
again, we turn to our resident expert (because he is a Ghost), Nick:
Nick turned away from the window and looked mournfully at Harry. “He won’t come back.” “Who?” “Sirius Black,” said Nick.
Then, we have
Lupin and Tonks, who both died in the final battles, and, like the aforementioned students in the last battle, died doing what they believed in.
The best candidates
Voldemort and Snape
also seem unlikely; the first because of too much damage to his mind and spirit, the second because of what he hoped awaited him in the afterlife. That being said, the first still seems like an excellent candidate, if those obstacles were not insurmountable -- but, of course, he would then simply be a ghost.
And, of course, you specified characters that we loved, which lets him out. For most people, anyway.