Anything of unknown breeding, or anything of breeding known to carry Frame, such as QH and Paint and minis, should be tested. Frame is in the TB world, but just a small handful of lines.
Is white or lack there of any indicator or is it simply breed specific?
Simply breed-specific in some cases (ie QH/Paint/minis) and anything with those mixes. A horse can be 100% solid, not a single white hair, and be a Frame carrier, so never rely on a visual for a solid or “just looks Tobiano” horse if the breed/breeding is potentially an issue
My filly has tons of white. If I were to breed her to the horse listed above (purely hypothetical to the Nth degree!) should she be tested or is the fact that her dam was a TB with little to no white anywhere in her family good enough?
Neither her TB side nor her WB side have any remote associated with Frame 
If someone were to pull a mare out of the kill pen with absolutely no inkling of her background but little to no white would you test her before breeding to the stallion above? I guess I am trying to judge just how big of an asshat the person referenced in the OP actually is.
Absolutely 100% test, because the QH influence is in SUCH a large % of the horse population. And when you add an obvious spotted pattern to the mix, the odds increase, since the odds of there being APHA blood becomes higher, and APHA has a much higher prevalence of Frame than the QH
[QUOTE=oldernewbie;6864338]Knowing what I know now, if I pulled a horse from the kill pen, I would have it tested, even to be a riding horse if it were a stock type. I’ve seen dumping first hand from paint breeders for HYPP, have heard second hand about dumping for Arab people - mostly just to get them off the feed bill or to cull, and I feel fairly confident that some QH folks do it as well, probably again for HYPP. And knowing that even solid colored horses can be carriers for lethal white…well, it would just be worth the money.
As I have a gelding, not actually an issue for me![/QUOTE]
Yep, I would have any kill pen or otherwise unpapered horse tested for HYPP at the very least, because if you end up with one, you AT LEAST need to know it’s there and do what you can to manage it, which means paying close attention to potassium intake and stress levels. No need to test a gelding for Frame.