I don't own any vintage hats so I had never considered this, but it doesn't sound like a bad idea. Even if someone says it was their father's or grandfather's hat and that it's been in a closet for several years, you/we don't know the hat's history and can't know what kind of harmful particulates--asbestos, for example--might have lodged in the felt over the years. Felt fibers might be the least of your concerns. But then, it makes sense that this advice would come from a Doctor.
And I have and use plenty of masks every day. [emoji2]I don't own any vintage hats so I had never considered this, but it doesn't sound like a bad idea. Even if someone says it was their father's or grandfather's hat and that it's been in a closet for several years, you/we don't know the hat's history and can't know what kind of harmful particulates--asbestos, for example--might have lodged in the felt over the years. Felt fibers might be the least of your concerns. But then, it makes sense that this advice would come from a Doctor.
It almost never works, in my experience. You end up replacing the ribbon and taking the brim binding off, or accepting the stains and calling them "character".
Alden 403 Moc Toe Boot - Brown Chromexcel - $680 The classic "Indy" style Alden Boot in the classic Horween brown Chromexcel. 
Grant Stone Diesel Boot Dark Olive Chromexcel - #395 Goodyear welted, Horween Chromexcel, classic good looks. 
Himel Bros. - The Ross Mk. 1 Leather Jacket Classic D-pocket motorcycle/aviator style jacket. 
Do the button end like a yo-yo string with your finger in the hole. Enlarge the hole out and slip it over the crown.
Came to me later...like a dog choker chain too.
I have not had much success with getting stains out of ribbon. Even in the naptha bath process the stains still remain. The felt cleans up nicely but the ribbons not so much.
I just retired after a 10 year career in industrial health and safety as an indoor air quality tech. I would suggest that unless you are sensitive to dust then the fibres/dust created by brushing your hat will not be harmful to your health. If you watch a custom hatter as they sand a felt body I have yet to see one don a mask. Both the volume and the exposure time are at the very low end of the hazardous exposure continuum that they do not present a danger unless as I stated you have a sensitivity or allergy to dust.
Yes, a goodly chunk of my work was what I called 'chasing ghosts'...that is investigating complaints of unhealthy indoor air quality made by somewhat paranoid employees that were convinced the air was killing them. The average PPM of dust in office buildings is in the 12 range. I would have fun with folks slapping my cotton work shirt next to the Particulate Metre and watch the readings jump into the mid 100's ....demonstrating that they carried more dust on their clothing than the surrounding environment. The funniest was a contractor's office with an employee complaining of allergic symptoms as soon as he arrived at work. I did a thorough investigation and as the office had likely never ever been cleaned or vacuumed the static dust levels were over the moon once disturbed. We presented the owner with a $2000 invoice and the recommendation they conduct a thorough house cleaning. The owner exploded..."you mean to tell me I spent $2000 for that a..hole and you are telling me the simple remedy is he needs to vacuum?"
