What it does depends on who you ask. I've heard about and seen a lot more bows that broke with dacron strings than with FF--but when a bow malfunctions with a dacron string, normally any and every thing but the string gets blamed. If someone stumps their toe with a FF type string on their bow, then it's the FF's fault.
Sometimes it can damage a bow, if the bow wasn't made for it. String grooves cut improperly and/or tips not reinforced are the two main factors I've been told about.
ALL materials have some stretch...some just have a lot less than others. I don't see lack of stretch as a negative, since you can normally take a bow that has punishing hand shock with a dacron string, put on a FF type string, and tame it right down. Less shock means less energy is resonating down the limbs--how can that be harder on the limbs?
I've been TRYING to get answers on why some bowyers say what they do about strings, with little to no luck--nobody wants to reply. One will void the warranty if you use a flemish string on his bows, or if you use dacron (NOT FF__DACRON!). Don't know how to contact this bowyer, had no luck with one of his dealers. Another claims flemish strings are "less accurate, dangerous, and could damage your bow". I e-mailed this one, no reply. Heard from yet another that claimed flemish strings can cause limbs to twist--didn't bother trying to get an explanation on that one, because it's ridiculous and false.
Anyhow, to answer the question...if the bow is made properly for the material, and the string is made properly, it doesn't do any damage...at least I've never seen or heard of verifiable proof. I use a FF type material (Dynaflight '97) on all my personal bows, including my selfbows. One longbow has been my "go to" bow for the past 15 years or so, it's never had anything other than a FF type string on it, and it's still going strong--no damage at all from the string, although it's back at the shop for it's third re-finish (I'm real hard on my bows--not on purpose, I just use them).
Chad