There are at least two ways to answer this question. One is to look at attempts to assimilate Native Americans. The other is to look at problems caused by actual, successful, assimilation.
To the degree that Indians have been successfully assimilated into mainstream American society, the main harm has been to their culture. Indians who successfully assimilated stopped being culturally Indian. The more that this happened, the more Indian culture as a whole disappeared. We could also argue that there was some harm to individual Native Americans who experienced guilt over having lost touch with their ancestral culture.
I would argue, however, that the greater harm to Indians came from failed attempts at assimilation. These attempts left many Indians stuck, as it were, between cultures. Beginning in the late 1800s, the US government attempted to assimilate Indians by destroying their native cultures. They did so through such things as the boarding school system and the Dawes Severalty Act. At the same time, however, they left Indians on reservations where they were cut off from white society. This was, arguably, the worst of both worlds. On the one hand, Indians were not actually made a part of American society and so they had a hard time feeling that they were truly American. On the other hand, their native culture was destroyed (or at least badly damaged) and so they could not be culturally Indian to the extent that their ancestors were. This left them as neither fish nor fowl, which is one factor that has helped lead to problems like alcoholism and suicide on Indian reservations.
Thus, Indians were harmed when they were successfully assimilated, but they were (at least in my view) harmed more gravely by attempts at assimilation that did not completely succeed.