I've come around to a solid "No" on this. I was a lean no, but am now a no - probably because 1) I grew up on football, and it never, ever reinforced a negative in my mind about Native Americans and 2) there is a small clique trying to force this change which has questionable standing. The difference between this and "Fighting Irish," for e.g., is also blurry to me.
Any and all stereotypes or slurs should be taken out of sports. Fighting Irish is a stereotype just as much as the Cleveland baseball franchise's logo (or now ex-logo) is a stereotype. The difference between Washington and, say, the Minnesota Vikings is that 1. all of the Vikings are dead, and 2. "Viking" is not a slur. Still, I don't believe solely a face should be a sports logo for whatever reason. If the Viking was displayed in a battle stance, then sure. Same with the Native American in the Washington logo. If it's about respect, make it about respect. Make the Native American appear as something to be revered, not mocked.