Does tapping Swiftfoot Boots remove Hexproof from the creature it's equipped to?
Tapping an artifact, or any other permanent, does not change its abilities, or whether those abilities have an effect, unless the ability says so. Tapped vs untapped is simply part of the status of a permanent [CR 110.6].

Historically, artifact abilities did in fact "turn off" when that artifact was tapped. Many of those cards have since been reworded to explicitly only work when the permanent is untapped. Examples include Winter Orb, Static Orb and Howling Mine.