There are plenty of examples, even for $C^{\ast}$-algebras. By the results of Connes and Haagerup, for $C^{\ast}$-algebras amenability is equivalent to nuclearity, so I will work with nuclearity, which is more familiar for operator algebraists.

For group $C^{\ast}$-algebras of discrete groups nuclearity is equivalent to amenability of the group. So, for example, the group $C^{\ast}$-algebra of the free group $F_n$ is not nuclear. On the other hand, it can be embedded in a nuclear $C^{\ast}$-algebra, as I will show below.

Whenever we have an action of a group $\Gamma$ on a compact space $X$, we can construct the (reduced) crossed product $C(X) \rtimes \Gamma$, which contains the group $C^{\ast}$-algebra $C^{\ast}_{r}(\Gamma)$ as a subalgebra. We now need an example of an action of a free group such that the crossed product is nuclear. We can use, for example, the action of the free group on its boundary. You can visualise the free group as a tree, using its Cayley graph, and the boundary in this case will be the boundary of this tree, i.e. space of infinite paths up to natural equivalence. This action is amenable, which is exactly a property needed for proving nuclearity of the crossed product. To sum up, the inclusion $C_{r}^{\ast}(F_n) \subset C(\partial F_n) \rtimes F_n$ gives you an example of a nonamenable subalgebra of an amenable algebra.