I have had this debate with myself, but for me it is choosing between a cloth mask and no mask while running. I have worn a cloth mask fine in the grocery store with no moisture issues, but I know if I were to wear it outside while running it would get wet from my breath, and there are studies that suggest wet cloth masks are actually worse for infections than no mask at all (for example here: https://bmjopen.bmj.com/content/5/4/e006577.short A 2015 study from Australia and New Zealand on cloth masks vs medical masks).
I live in an area where I can usually keep 6 ft away from people while running (usually a lot more than 6 ft because April has been rainy and cold and that is great for my running), so I am fine with no mask while running.
Long term effects of breathing through a gas mask while running though... I can imagine that might lead to some serious face chafing issues, which might be lessened with anti-chafe products like Body Glide.
If less air is getting in, you will be working your lungs and your heart a whole lot harder than they are used to. I've run while mildly to moderately anemic over the past year due to health issues, and it is a whole lot harder to run when you can't deliver enough oxygen to your muscles. You might experience chest pains or dizziness, especially going up hills. Chest pains due to too little oxygen in your blood is "not great", from my own experience, and can be very scary. How much less oxygen are you getting with the mask? Is it something you think would be healthy to work your heart and lungs a little harder, or do you think that might put too much pressure on them at a time when you want them to stay as healthy as possible?
I'm not sure if there are health risks for breathing in and out of a very moist environment like the inside of a gas mask, humid air is harder to breathe and can be a problem for people with asthma. A hot and moist environment is prime breeding ground for bacteria, so you would probably need to clean it out well and often.
You also might want to consider what the benefits are of wearing a gas mask versus finding an alternative. If the main goal is to prevent spread of the virus and you can have some sort of mask that you can put on while passing people and then put in a pocket when not around people, that might be a better option for your long term health.
I hope you find something that works for you and that you stay safe and healthy!
I wouldn't do it if you're running outdoors simply because you don't need to so why bother? Run somewhere where there aren't a lot of people around and stay six feet away from everyone else. I do my outdoor exercise at night, when there are few out and so no problem. But if you are completely healthy, yes, I suppose you can, but if you're wearing a full gas mask rather than a simple cloth mask or dust filtering mask it is going to be bulky and heavy and, of course, unless you got one fitted for you the truth is, almost none of the masks we're using are all that effective if you're not staying six feet away from everyone anyway. Those respirator masks docs wear are supposed to be fitted to them, but of course due to scarcity they're not anymore. You will get less oxygen so you might get dizzy or lightheaded, and the tighter and more elaborate the mask the less oxygen you're going to get. You're also going to be breathing back in the carbon dioxide you're expelling. If you're going back to the gym, you're not probably going to have any choice in the matter, and over time unless you have a respiratory condition of some kind your lungs and heart might get stronger, but you'll also be working them a lot harder so if you run the same distance and it's a long distance that you're used to it will be equivalent to running longer, and some evidence shows that although in the short term running improves our hearts if you run say marathon distances there are more cardiovascular incidents because the body can only take so much. You never know until bad stuff happens if you're one of the unlucky ones. So yeah, you can do it. But if you're outdoors, I don't personally see why you would unless where you are is requiring a mask even for outdoor exercise. It's up to you, just be aware of how you're feeling until you get stronger. What I don't know is if anyone has ever studied the long-term effects of heavy exercise wearing a mask since nobody has ever worn them long-term before. Again, we have to be breathing back in some carbon dioxide and I just wonder if that's a thing or not, I don't know. Peace.