on burgers or hot dogs - honey mustard
on french fries - tartar sauce or honey mustard
pesto sauce (fortunately many of the pizza places where I live have pesto as an option, though I usually have to still specify NO RED SAUCE or they just do both). I actually didn't eat pizza at all for nearly 5 years before I figured this option out.
muhammara - this is a combination of walnuts, roasted red peppers, and pomegranite molasses that has worked well in a couple of different recipes we've made that called for tomato paste. Not sure how widely it would work. Not a good option for people who can't tolerate any nightshades, obviously.
mangoes. Ok, so I haven't actually tried this yet, but I have seen a salsa made with mangoes and no tomatoes. It sure sounds good!
That's all I'm coming up with off the top of my head, though I'm sure there are more. In a lot of things they end up being not a key ingredient and can just be omitted with no substitute.