Examples for lower bounds proof except sorting
Here are some examples:


Finding an element in a sorted list takes time $\Omega(\log n)$ in the RAM model.
Implementing a disjoint sets data structure requires an amortized $\Omega(\alpha(n))$ operations in the cell probe model.
Determining whether the input string is a palindrome requires $\Omega(n^2)$ operations on a single tape Turing machine.


There are many others.