Before the arrival of Europeans, Native Americans in the New World had no knowledge of Christianity, just as Europeans had no knowledge of Native American religions. To what extent did contrasting religious beliefs and assumptions influence relations among Europeans and Native Americans in the New World in the sixteenth century?

This is a huge question, but the short answer is that it would be difficult to find two types of religion more destined to clash (here, I will be lumping together all the Native American religions with the broadest strokes for the purposes of discussion, which is not to minimize their nuance and individuality).
First, Native American religious practice was joined with everyday life: there was no separation of the sacred and the secular. In Christianity, there is a divide between the sacred and the secular, with specific times and places of worship that exist apart from everyday life. This led to confusion for the Native Americans, especially since most of the land they lived on was considered sacred space. Where they farmed was sacred, for example, and they rejected the European concept that this could be taken from them or swapped out with some other piece of land. The colonists and missionaries, for their part, couldn't understand that seemingly secular practices, such as dances, were sacred to the Native Americans. To the missionaries, the Native Americans could seem like irreligious heathen because of their merging of secular and sacred.
Further, some of the symbols that tribes worshipped, such as snakes, were traditionally considered Satanic by the Christians, and so some tribes were wrongly labelled devil worshippers.
In addition, many tribes practiced henotheism, meaning they believed that different gods protected different territories. This led them to embrace polytheism, or the idea that there is more than one god. The Christians strictly rejected polytheism. Native Americans could not understand why the Christian God and other gods couldn't coexist, while the Christians were bent on converting the natives and eradicating what they thought were their false gods or idols. Christian exclusivity thus clashed with Native American inclusivity.
These religious differences needed much space for talk and a context of mutual respect for the different cultures to stand a chance of understanding one another, but that kind of dialogue was in very short supply.

One of the major factors that contributed to colonization and imperialism by Europeans of people in the Americas and elsewhere was the ethnocentric mindset held by Europeans. Many Europeans held the belief that the Native Americans were, at the very least, inferior. In fact, many Europeans didn't just believe that the cultural and religious beliefs of Native Americans were inferior, but rather believed that the Natives had no culture or religion at all. This led to the belief by Europeans that they were not civilized peoples, which helped to justify the conquering of Natives and the taking of their lands. Additionally, many European nations not only attempted to gain control of resources in the Americas, but also competed to "save" Natives by sending Christian missionaries to convert them to Christianity and to introduce them to European values and culture.
http://pluralism.org/encounter/historical-perspectives/first-encounters-native-americans-and-christians/

Contrasting religious beliefs and the mutual misunderstanding of each others' religions affected the relationship between Native Americans and Europeans in the 16th century and beyond. For example, English colonists (who came to the New World in the late 16th century and afterward) thought of the Native Americans they encountered as satanic and as deserving of displacement from their ancestral lands. In Jamestown, for example, the Powhatan followed a polytheistic religion with many spirits to whom they made scarifies. The English settlers, rather than seeking to truly understand the Powhatan religion, tried to displace the Native Americans from their lands. Later, the English settlers attempt to convert the Native Americans. There was no attempt at syncretism, or combining the two religious traditions. In addition, the English misunderstood a great deal of the Powhatans' religious practice; for example, they thought that the male initiation ceremony involved sacrifice, which it likely did not.
The Spaniards sought to convert the Native Americans they encountered to Catholicism. They also sought to use Native American labor on their plantations, called encomiendas, and, in return for the labor, the Spaniards promised to save the Native Americans' souls. The Native Americans often believed in communal land ownership, and they did not always understand the Europeans' practice of private land ownership at first; the Natives were often displaced because of the Europeans' desire for land.