Why is it that after workouts
 

If I lie on my stomach, my ears fill with a humming sound and I start to get a headache?

Let me guess : Don't lie on your stomach after a workout, then!

:)

Seriously, anyone else get this? Once my heart rate is down, it's fine. Same thing happens in reverse : if I'm doing a floor workout and stand up, it happens too.

It could be a difference in blood pressure when you change your elevation. Or the blood rushing from prone to upright. Try getting up or down slowly.