You're going to get different answers depending on which nation you look at. I'm a bit short on time right now, so I'll focus on just one: the Haudenosaunee (Iroquois).

Initially, the official position of the Grand Council on the brewing conflict between Britain and the colonists was one of neutrality. The looming war was viewed as an internal squabble among the Euro-Americans and not something that concerned the Haudenosaunee. However, within the Six Nations that composed the Haudenosaunee, factions were forming that favored alliances with one side or the other.

While the hereditary leadership of the Oneida and the Tuscarora favored continued neutrality or leaned toward the British, their warriors were increasingly in favor of supporting the revolutionaries. This was in part because of their close proximity to colonial populations and their recent deals that had occurred since the mid-1760s that helps secure a tenuous border between New York and the eastern portion of the Haudenosaunee. The influence of the charismatic Calvinist missionary Samuel Kirkland also helped sway many of his recent converts to the revolutionary side in those two nations.

Meanwhile, the Mohawk tended to favor the British. They had a good relationship with the British, especially thanks to Thayendanegea (aka Joseph Brant). They also relied upon recent agreements to secure their eastern border with New York, the Mohawk council saw value in having the British government around to keep the colonists in line and enforce prior agreements. The British might be land-hungry imperialists, but having them around was the lesser of two evils compared to a bunch of colonists doing running amok.

The remaining nations of the Haudenosaunee (the Onondaga, Cayuga, and Seneca) were the main forces in favor of neutrality, with slight British leanings (again, better the devil you know). In 1777, once the war began, the Grand Council convened to discuss the matter. When the Council failed to reach a decision, the various factions pulled the Haudenosaunee apart.

Fearing that the Mohawk would side with the British, the colonials attacked their towns along the Mohawk Valley, displacing the population to Montreal, Fort Niagara, and a refugee settlement in Oneida known as Oquaga. Mohawk property was seized, not only by the colonial militias, but also by pro-colonial Oneida who had officially joined forces with them at the time, along with the Tuscarora and the Stockbridge Indians, a community of Lenape and Mahican that established the town of Stockbridge, MA. A brief side note: the Stockbridge Indians, in particular, got the short end of the historical stick. Many returned from fighting on behalf of the revolution to discover that in their absence, the colonists had taken over Stockbridge. The Stockbridge Indians ended up largely moving onto Oneida land in New York after the war.

While a sort of civil war was being fought between the British-allied Mohawk and the revolutionary-allied Oneida and Tuscarora, the other three nations hunkered down and attempted to maintain their neutrality. Officially they had not taken sides. Certainly some of their men were already fighting, but that was their individual prerogative not public policy. That changed in 1779.

Again fearing that the western Haudenosaunee would soon be joining the side of the British, Washington ordered a series of preemptive assaults. The first was commanded by General Clinton against the Onondaga in April. In June, General Sullivan led another assault from the southeast to ravage the countryside of the Cayuga and Seneca. Colonel Brodhead led a third expedition from Fort Pitt, heading northeast into Seneca territory to met up with the others already in New York. There was only one major battle between colonial and Iroquoian forces during this (the Battle of Newtown on August 29th). While the revolutionaries won the Battle of Newtown, it did but the Haudenosaunee non-combatants more time to evacuate. Despite devastating a considerable amount of the Haudenosaunee's growing crops and causing mass starvation that winter, the Sullivan Expedition failed in its goal of capturing and detaining a any significant portion of the population.

Though Thayendanegea and others (including many Oneida and Tuscarora who began to switch to the British side leading up to the Sullivan Expedition) would continue to mount raids against the colonists, the Haudenosaunee as a major political and military power was knocked out of the fight.

The Grand Council fire, which had been put out following the debate on the which side, if any, to support in the war, was thankfully re-lit once peace had been re-establish. However, it was no longer a single fire. As the Haudenosaunee began rebuilding after the war, two successors and one offshot all lit their own council fires. The New York League was now based in Buffalo Creek rather than Onondaga. Thayendanegea helped establish the Canadian League (The Six Nations of Grand River) in Ontario. Both Leagues operated independently, and rebuilt the hereditary portion of the Grand Council for their own lineages (I should mention that Thayendanegea was not a member of the hereditary council members, but was a "Pine Tree Chief" - an individual who received a seat on the council due to noteworthy national service).

In what's now Ohio, a third league formed. It included Seneca and Cayuga, but was actually spearheaded by the Wyandot and included non-Iroquoian peoples as well. The Ohio or Western Confederacy sent a letter to the US Congress after Britain and the US declared peace, seeking to convene a peace summit of their own. However, this invitation was ignored, as the US believed it had inherited Britain's claim over all land east of the Mississippi and did not recognize the Western Confederacy as a sovereign entity. Because of this, from the perspective of those living west of the Appalachian Mountains, the war that had begun in 1775 didn't end until 1795. But that's another story.