Nope, my puppy doesn't tear that I've noticed but he did and now that you mention that they tear when they teethe, it makes sense now.
The one with the tear stains is a year old. I haven't noticed that his eyes are tearing but he had to have gotten the stains from somewhere.
BTW, acv is also good for ear wash and yeast infections in the ear. It has tons of benefits for your dog (and YOU if you can stomach it)