If I'm being honest here, I've been doing light runs for about a month now, and I always wear my binder. I have two underworks binders, so I normally just wear the older one which is slightly looser. I haven't actually noticed it being much of a problem, though it certainly collects a lot of sweat, and to a lesser extent, heat.
I probably shouldn't be doing this, but my dysphoria is so strong any kind of exercise is hard for me, what with my clothes sticking to my body, people being able to see me, etc. that I couldn't do it at all without wearing the binder. My suggestion to you is either loose binder or a very tight sports bra/several sports bras. The one LearnedHand suggested looks like it would work fairly well.