Motion/ forces in different directions can indeed mix to some extent and make for some very complicated and bizarre effects. However a force that's applied truly in a horizontal direction will have no effect on the vertical forces, and in balancing them you should consider horizontal and vertical as separate directions.
The real world is never so perfect, so it's hard to say in practice. However I expect that a runner isn't so much exerting less pressure on the ground; rather, he is spending less time exerting that pressure. Overall this might give an appearance that he'd be lighter if weighed, but he isn't really. Since you couldn't actually weigh him in any meaningful sense while he's running it's a bit hard to confirm this. And also possibly the effect of wind flow might even make your runner appear heavier. Or lighter, depending on the dynamics of the thing.
Planes can fly because in some sense they weight as much as air does, when flying (at least the overall downward force is much less than the true weight). Formula One cars are designed to increase the downforce, which amounts to making it seem like the car weighs more when it's moving (increases the stability when going round corners, I think).
TLDR; it depends. Real motion is complicated.