Is it a problem if my dentist often finds a lot of tartar built up? i.e. is there any harm given they just remove it each time?

No Harm: More harm to the gums will be done if they keep the tater. Remember that the tarter will provide a place for the bacteria to grow and prosper causing not only gum disease but affecting you overall health. Just Do it

Damage: Tarter damages your mouth 2 ways. 1). It acts as a platform to support acids and digestive enzymes, both of which cause gum/bone recession. 2). It acts as a mechanical irritant, causing bone/gum recession. Best thing you can do is prevent Tarter build-up. Talk w your Dentist about Tarter prevention oral hygiene techniques, or ask for referral to a Periodontist.