Great question! As you probably know, there are different forms of mimicry. For example, a moth may look like a wasp. There is also mimicry like a bird copying songs that it hears.

For the first kind of mimicry, animals have to have a certain appearance, which is obviously innate. They also have to have behavior that matches their appearance. It doesn't help much to look like something if you dont behave in a way that adds to the disguise. For example, if you're a caterpillar disguised as a bird dropping, you cant be moving around during the day because you wont fool anyone. In all of the cases of mimicry disguises I can think of, the behavior seems to be innate. Of course, a behavior can be both. That is, an animal can have a basic program for a behavior that it fine tunes with experience, but getting eaten doesnt leave much room for learning to do better.

The broken-rays mussel. It has an extension of its body that looks and behaves a lot like a fish:

We know for certain that it didnt learn to make those movements by watching fish because it doesnt even have eyes! Another way we can tell whether a behavior is innate is whether animals do it right after they hatch, are born, etc. because they havent had time to learn anything.

The second kind of mimics are the ones who imitate sounds as part of their own songs. They clearly learn the specific noises. Heres a great video of a superb lyrebird imitating construction noises:

The ability to imitate and the behavior of doing so are innate. Scientists think that this sort of imitation allows the male to advertise to the female that he is old enough to have learned a lot and that he is in such good shape that he can spend a lot of time singing. This may convince the female that she should mate with him.

What are some of the pros and cons of learning vs. innate behavior? Why arent we born already programmed with a lot of skills? Why cant a mouse learn as much as a person can? If questions like these interest you, you may want to study animal behavior.