Broadly speaking, the symbolism being used is usually either an obvious caricature of the real thing (see: the former Washington Redskins), or uses cultural and religious elements that are at best kind of offensive in the context of being used as a sportsball logo (see: most uses of peace pipe or war bonnet imagery).

There are exceptions, but they're of the kind that generally come with either somebody actually doing the research or people of the given group actually participating locally.

As someone else mentioned most of the push against is not from American Indians/Native Americans but vicariously offended others.