I'm trying to design a mammal that lives in trees. I'd like its hands to be good for climbing, but not good for using tools.
What aspects of the hand would make it good for climbing and not good for tool use?
Lots of creatures climb, few creatures create and regularly use tools. Fewer still if you want human level dexterity in the hands.
So the common themes here are.
No opposable thumbs. This makes tool use more difficult and would increase wear on a tool, a firm grip is key for not stripping a screwdriver head for example.
Claws. They are sharp. They break things.
The first thing that come to my mind is a Sloth. Just think of trying to use any tool with claws 4 inches long. Then think of trying to climb with grappling hooks!
Their specialised hands and feet have long, curved claws to allow them to hang upside down from branches without effort. They usually eat, sleep, and even give birth hanging from limbs. They sometimes remain hanging from branches after death even if shot from below by humans.
All you have to do is look into nature.
Geckos use setae in order to stick to surfaces, and if your hands have setae it would allow the creature to climb smooth surfaces easily (perhaps really large trees with smooth bark?).
But climbing can't just be generalized to smooth surfaces, and you mention trees - so what other aspects can help improve this? Well, snakes can climb trees. Snakes get their friction from their scales, which could also damage tools, so this counts for double points!
But whenever I think of climbing trees, I think of squirrels. Have you ever noticed how damn fast they move up them? I'd use their paw design - 4 longish fingers with claws on the end, with no opposable thumbs. Turns out, opposable thumbs are pretty important for using tools.
The verdict? Use a combination of natures finest. Style your mammals hands after squirrel paws, but also give them some scales and perhaps setae on the finger tips.
The ability to use tools effectively comes from the brain's ability to adopt this dead unfeeling extension as part of its body image.
A paw designed for swinging from branches or climbing or otherwise grasping in general might seem to make a perfectly good hand: if it can grasp a branch and hang on with enough force to support his own body weight, certainly then he can grasp the handle of an axe, right?
But, even with the grasping ability, he can't actually use an axe because his brain doesn't have the circuitry to make it function as an extension of his arm. It takes a great deal of practice to hit anything and the skill only works on that "trick" and doesn't generalize.
Note that I'm not referring to other aspects of intelligence. These creatures many have complex social interactions with politics and schemes, remember vast territories of resources and what signs indicate what to expect of the world around them. But using tools is a particular brain feature, and one we take for granted.
Bad for tools? Just take away the thumbs.
Good for climbing is very open-ended, as the other answers have showed with plenty examples. Here is a fun one:
Hands with lots of tiny retractable claws all over the palms
You put your hand against vertical rock, and any cracks will get filled with your little palm-claws, giving you heaps of purchase. These claws would give a world of pain when you slap someone, too!
A downside is a slimy surface would be impossible to climb, but you can use that as part of the ups/downs of the system.