This also happens to me, and while it has never been overly bothersome, I have always wondered why, so thank you for asking this question. Here are some things that I hypothesized...
1. Maybe exercise induces the release of metal ions into the saliva, causing you to taste that oh-so metallic taste. Turns out there is no evidence of this...
2. Maybe increased blood flow to the tongue causes the taste-buds to be hypersensitive to tastes already present... also no evidence to support this.
So as I did my research, I found many websites that talk about Exercise Induced Anaphylaxis. I have also been curious about this. It turns out one of the most common causes of metallic taste in the mouth is that of a mild allergic reaction. This is seen sometimes in response to exercise, your body can begin to have an allergic reaction to it. It turns out a brother-in-law of mine had a rather serious allergic reaction in response to the heavy workout that my brother and he went through. Hives, dizziness, swelling, everything.
Ironic, eh, that some people can be allergic to exercise? But that isn't a reason to stop exercising just because you have a metallic taste in your mouth, I also get this when I hold coins in my sweaty little palm for more than a minute or so, and also whenever I grasp a barbell, I start to get the same taste, which I suspect to be related to me touching metal and either having absorbed it, or having rubbed my face somewhere along the way and getting a whiff of that to my olfactory or gustatory
senses... Anyways, exercise induced allergic reactions may be the reason why. It is more likely after a heavy workout (such as climbing the Y, or running). As far as the hurting lungs, I would most likely attribute this to hyperinflation due to the exercise induced hyperventilation and trapped air when people don't take enough time to breathe out/exhale.
Please remember that if this condition bothers you in any way, a physician is the only help or advice you should heed, as I am not a physician.
Thanks for asking this interesting question
Have Fun Storming the Castle,
-Il Guanaco, who may at sometime in his life say "I'm allergic to exercise" and laugh knowingly.
PS. I am still waiting on some professors to get back to me on the sources they think this answer might come from, so if you want them, email me at il(dot)guanaco(at)gmail.com