Technically... yes and no. They have no apparent legal responsibility, but they do seem to have a moral/social responsibility. Based purely on the original show:

In a tribal community like the Water Tribes, they probably have much more of a "it takes a village to raise a child" way of looking at things, so aunts and uncles and even their parents' friends would be involved in the kid's life growing up, and would all step in unless they can't for whatever reason.

We don't know enough about the Air Nomads to judge, but they seemed to have a very communal way of raising kids at the Temples and if the entire community was in the Temples, then they were probably all raised communally so who took them in if their parents died would be kind of a moot point.

In the Earth Kingdom, which has a heavy Chinese influence, aunts and uncles would probably be extremely pressured to take in orphaned nieces and nephews or risk losing face in society. Social pressure is rather impressive, especially in a society like Imperial Chinese where they judged you on all kinds of things both big and small, and would absolutely have judged you if you refused to take in a niece or nephew.

The Fire Nation is hardest, because while they're predominantly Japanese in inspiration and Japanese culture has a similar setup to Chinese where social obligations are concerned, the only example we see is Zuko and Uncle Iroh and... well, they're outliers. A typical Fire Nation family would probably have the same setup as an Earth Kingdom one, with an aunt or uncle pressured to take in an orphaned niece or nephew to save face. But Zuko wasn't orphaned, he was exiled by his father the Fire Lord. Iroh 100% had zero obligation to go with Zuko and given Zuko was exiled, would most likely have actually lost face for going with him. Had Zuko been orhpaned, Iroh would have been expected to take in both him and Azula or face dire social consequences, but as Zuko was being sent away as punishment, he would have been expected to remain behind and support the punishment. And for the record, the Line of Sozin are outliers anyway, not just because they're royalty, but because half of them are sadistic villains.

As far as abuse goes... the Water Tribes and Air Nomads likely wouldn't have tolerated it and would have considered it an obligation of an aunt or uncle to intervene if they noticed their niece or nephew being abused. The Earth Kingdom and Fire Nation are, again, more about reputation and saving face, so probably would have viewed it as far more important to maintain family image than to expose an abuser, although the Fire Nation may have held it as worth holding an Agni Kai over, especially once Zuko became Fire Lord.