There were three main groups of non-Christians who partook of 18th century American society: enslaved Africans (of whose religious practices I know very little), Native Americans, and Jews. Native Americans typically received Christian baptisms as a prerequisite to joining colonial societies after choosing (or not choosing) to opt out of a Native-controlled setting. The best examples of this are the "praying towns" in 17th and 18th century New England and the réserves along the St. Lawrence River in New France. They still, of course, faced racial discrimination, but the religious discrimination these Natives face typically boiled down to doubting the sincerity of their conversions (less of a problem in some places than others).

Jewish people experienced discrimination across the entire continent. New France and Louisiana banned Jewish migration outright. Religious toleration in general was usually qualified with a clause saying something to the effect of "except Jews and Catholics" in British colonies. There are many, many episodes of Jewish discrimination including banning synagogues or just general anti-Semitism, but New Netherland (later New York) has an interesting history in that many of these legal barriers were stricken down.

The Dutch founded New Netherland as a trade outpost for procuring furs from the Iroquois in exchange for guns and manufactured goods. But even though Dutch Republic allowed Jewish residency and commerce in the Netherlands, New Netherland was envisioned as an entirely-Dutch enterprise. That meant Dutch migrants and only Dutch Protestantism. The governors were appointed with a mandate to preserve this Dutch-ness.

But Jews were involved in Dutch colonialism outside of New Netherland. They were essential to Dutch claims to Brazil, as they had established a Dutch-jewish outpost with which to compete with the Portuguese for control of the region. The Dutch lost in 1652, rendering the Dutch Jews homeless and vulnerable. They began relocating to New Netherland in 1655 to a mixed reception.

Director-General Peter Stuyvesant and Domine Johannes Megapolensis protested angrily about this influx of Jews. They had already been forced to compromise their vision for a Dutch-only colony by allowing English Protestants and Swedes to settle in order to bolster the colony's very low population. They had tried to prevent thee migrations or even expel them, but their Dutch superiors ordered Stuyvesant to allow them to stay, citing Dutch principles of tolerance and a need for a larger population.

The Dutch lords responded in 1655 that any attempt to ban Jewish migration would be "unreasonable and unfair" due to their vulnerability following their forced migration from Brazil. They also cited how important Jewish investments were among the capital of Dutch West India Company. So Jews were allowed to stay, but still faced legal discrimination. As in the Netherlands, they were not allowed to operate open-to-the-public stores, though they could still make sales in an unofficial capacity as private salesmen. Neither were they allowed to open synagogues under Stuyvesant's tenure, which lasted until England conquered the colony in 1664. They were, however, specifically allowed to trade with the Natives at Fort Orange.

These barriers weakened over time. By 1657 Dutch Jews were operating open-front stores and were allowed to continue despite grumblings from Stuyvesant and Dutch pastors. Stuyvesant even capitulated in allowing the Dutch Jews citizenship in 1658, which gave them commercial freedom on par with a Dutch Protestant. The prohibition on open practice of Judaism via synagogues remained, but Jews were allowed to practice in private ceremonies within private homes.

Edit: I can't believe I forgot to mention enslaved African as non-Christians. I don't really know enough of African religious practices in North America to comment on them, unfortunately. I changed the beginning so I don't implicitly exclude enslaved Africans.

Sources:

James Axtell - The Invasion Within

George Procter-Smith - Religion and Trade in New Netherland

I'll see if I can get back to this later to expand on it, but in the meantime, in a past answer here I briefly discussed this in terms of how Jews were treated in British colonies (in addition to what u/DarthNetflix wrote about New Netherlands and French colonies)-

Bear in mind- being in the British colonies didn't necessarily mean escaping religious persecution, depending on how one defines the term. It could mean taking it on. That first Jewish community in New York technically didn't have the legal right to worship in public until about 1700 (long after the right had been granted in England), and it was implicit that they were tolerated for as long as they kept their heads to the ground. (That first synagogue established at that time, Shearith Israel, still exists today.) Until the United States were established (and even, to an extent, after that), each colony had its own policy on acceptance of the Jews, from Rhode Island (which was entirely tolerant) to Maryland (where doubting the Trinity was a capital crime on the third offense). Even in places where Jews had the right to worship, though, they didn't necessarily have equal rights under the law- for periods of time in New York and Jamaica, Jews could not vote, for example. Some colonies retained test acts- whereby only men who affirmed faith in Jesus could attain public office- until long after they'd become states.

Jews also came to Spanish-ruled American colonies from Spain in order to escape persecution, and were actually successful for a while, until the Inquisition was brought across the Atlantic- for more about this, see my past answer here.