If“If you are so smart why aren't you rich? If” “If this country is so bad why don't you leave? If” “If it is so easy why aren't you doing it?” These types of rhetorical barbs rely on what is called an enthymeme, and argument with implicit parts. If X then why not Y? relies on Y not being in evidence, and suggests that X must not be the case. Enthymemes can be valid, when reconstructed, but if there is an error it is typically in presuming that X implies Y.
This case certainly is a non-sequitur, but that is a very broad category. Anything invalid is a non-sequitur. But why is it invalid? The reconstructed argument would be something like this: white privilege is the only way to benefit from race; Warren sought such benefits by claiming to be Indian, not white; hence, there is no white privilege.
But, of course, there is more than one way to benefit from race, by being white and by being non-white. Overlooking additional possibilities is generally known as a false dilemma. In this case, it is an even more specific kind of overlooking: overlooking (or ignoring) additional causes for the observed action. This is known as the fallacy of a single cause, or causal oversimplification. Indeed, this is the most common explanation for If X then why not Y?, when it is a genuine question, and the reason for it being a fallacy when not X is the implied conclusion.