CT nuts are flinging the crazy around already. Stupid false things I've seen promoted on social media just this afternoon, without making much effort:
Harris's parent were not citizens at the time of her birth (true) so she can't be President (false). If she can't be president, she can't be Vice-President (moot, since she can be president). (Jacob Wohl is pushing this one, Snopes has a good write-up debunking him).
Harris is not eligible to be President nor Vice President because she was born with dual citizenship and therefore does not qualify as "Natural Born" (false, she is still eligible, she lived most of her life in America after being born here).
Her father claims to be descended from a major slave owner (true, he did claim this). I think they are suggesting that this reflects badly upon her, or suggests that she can't actually be black? (Snopes has a good write up on this one). Apparently is is bad for her to have this ancestry even if it is not bad for many white Americans to also be descended from slave owners. Or maybe it is hypocritical of her, even if she has nothing good to ever say about that ancestor of hers (Which isn't fully proven, the record is a bit muddled).
Claims that she can't be black because her father was Jamaican, not Black (as if nobody in Jamaica has African ancestry, or is descended from slaves). The racial makeup of people in Jamaica seems to be a confusing issue for some people. (If she's not black, then neither is Colin Powell, who also has Afro-Caribbean ancestry).
Accusation that it is racist to refer to her mother's heritage as "Indian" because Native Americans don't like that word.

(Many native Americans don't like that word when it is applied to Native Americans, I don't think they have any problem with the word applied to people in or from India. In fairness, I once had a coworker who shared this belief, and she was as liberal as get all. Also bat-guano crazy.)
Mass confusion as to whether Harris can or should identify as more than one race. Mixed-race ancestry as it relates to personal identification seems to be very, very confusing to some people. She has Asian ancestry and she also has Afro-Caribbean ancestry and she can claim whatever combination of that she feels is appropriate. This is not complicated.
I am sure more will come up, it will be like Birtherism 2.0, only in a lite "diet" formula because she is only running for VP. With luck, this will make the racism of the conspiracy people, as well as their prominence within the Republican party wildly visible, such that the boot-licking apologists can't waive it away.