Yes, you should wash your face before and after exercise, otherwise you may get acne on your face. You may bring cleansing facial wipes with you if running water is not available.

I only wash my face after a workout. I work out at night, but I'll just work out and cleanse my face and apply BP, moisturizer, AHA as I do usually. If I switch it up and work out in the morning, I just rinse my face with water, workout, then go home and wash & apply treatment. This method has worked at keeping me clear thus far.