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Our teeth are prone to fractures when dry due to their crystalline internal structure. Because of this, a majority of terrestrial animals with a similar tooth makeup have lips to keep the teeth moist.

What composites for a tooth could be made to circumvent this issue? Ideally, this structure wouldn't be too metabolically expensive, and horns, claws and potentially fur could be made with some of the materials.

Teeth can regenerate. Even humans have two sets of teeth: temporary and permanent. Having a constant regeneration cycle will allow for normal teeth to survive without lips to moisturize.

But tusk bearing animals have them for a pretty long time, elephants have them from 1 year old until they die at around 60.

The tusk's ivory is maintained by having blood circulation throughout internals of the tusk. The only real metabolic cost is the maintenance of the tissue and water evaporation.

You can have normal teeth, just cover it with something other than lips. I'll propose something that will look aesthetically like bare tooth exposure, but I will cheat a little bit, so be free to down vote if you must.

Have them be covered on thin layers of keratin cells, just like the epidermis. Underneath the epidermis, allow a system like the lymphatics that will keep the whole tooth system hydrated. The moisturizer should be the same chemicals as saliva.

Now the problem, every time your creature chews something, this layer will be destroyed. As it gets shredded it'll release more saliva on the teeth, as the saliva vessels will be open, contributing to a better digestion and swallowing, but more importantly, a continuous stream of moisture.

If you have an herbivore, the rate which the layers must be repaired needs to be faster, as they're always chewing. That will not be too energy consuming since there's not much surface to cover. If you have a carnivore, they might get days without eating and the regeneration could be slower. You'll have to calibrate that system. Also, iguanas don't chew that much, they basically bite and swallow, keeping the layers more or less intact.

In order to properly answer this question, you have to first consider the process(es) put into place to allow this.

It would seem evident that it's simply much easier and more efficient to moisturize the teeth with the lips, and retain that moisture. Considering that teeth are a fundamental part of ones health. The energy cycle would need to implement this change in dynamic if other energy was being used to reconstruct, or to protect the teeth.

In the instance of the lips, the saliva is a somewhat double-medium and the lips offer multiple functions.

So, without the lips - you actually have more issues as well that the lips help simplify. Such as disease, breathing in of air from a polluted climate, and simply not being able to really control what goes inside of your mouth.

So really, while this is the problem you're trying to solve - this problem is relative within the nature of the true solution that lips provide.

In short, it may be simplest to simply remove the teeth altogether and grind your food in a morter in order to consume it.

Or,

If you could somehow integrate iron production within the body, and produce a form of titanium/steel - that would be the ultimate solution probably.

But really, this question is much bigger than simply caring for the teeth - because the lips offer many health benefits and evolution always finds the best fit.

As comments and another answer have put it, do tusks.

Or use cheratin like a Rhino's horn or bird beaks, or chitin like spider fangs.