When out of combat for 5 seconds, Boots of Mobility grant +5 move speed. What counts as being in combat?
Anything that causes damage or a debuf will count as combat. This includes damage you deal and damage dealt to you. So if you exhaust someone then your buff will wear off. As the Wiki states, "combat dehancers" do not count as direct combat.
A combat dehancer would be something like Warwicks Blood Scent. It reveals you on the map, but does not do any damage or have any other direct negative affects to your character.
As of V22.214.171.124 these no longer remove your buff from the boots.
The boots as an item are best used for roamers who need to switch lanes quickly. They are great for ganking because by the time someone calls an MIA you're already in another lane. Shaco is one of the most common champions I see with Boots of Mobility.
Check the Patch History for the item at the bottom of this page.