Are there any tree-dwelling mammals which might be found taking a living leech and eating it?
(And do any hollow-dwelling birds of prey eat leeches?)
First I'd like to point out that there are over 500 species of leeches. Most of them live in fresh water (nearly 90%), and so most tree-living animals would not come into contact with these leeches. Because of this a leech's biggest predator would be fish.
According to The New World Encyclopaedia the predators of leeches are fish, crayfish, turtles, birds, frogs, dragonflies, and damselflies. It states that many of their predators have also been their prey. Ecospark also adds snails and other insects to the list. Biokids also points out that leeches will eat other leeches.
Britannica cautions humans about swimming and drinking water with leeches in it. They can attach to internal cavities - such as the nose and throat and then the leeches can drop into the lungs and cause suffocation. In Asia, it is common for a domesticated animal to die in this fashion.
According to Animals.mom.me monkeys and other omnivorous animals will eat leeches that live on land, but land leeches aren't very common. After over an hour of searching various websites and information about leeches, there is no other source stating that mammals eat leeches.
However, it is possible to eat leeches. They did it on a television show called Fear Factor (video of the leeches being eaten linked). Before you go eating leeches though, remember these people were monitored both during and after ingestion of the leeches.
Short answer: There are no recorded mammals as predators to leeches, but it is known that pets have eaten leeches (as well as breathed them in). And humans are also able to eat leeches. It's all about if the leech is within grabbing distance when the mammal is hungry.