Look here for lower bound of time for matrix multiplication.
And here for the following result.
We prove lower bounds of order n log n for both the problem to multiply polynomials of degree n, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers.
However there is no any strong theory about lower bounds of time, there is one easy not-too-general way to get it. You can get some problem with known lower bound time and reduce it in linear time to your problem. For example speaking about heaps and operations “insert”, “erase maximum” and “get maximum” we get that at least one of these operations should have amortized time $\Omega(\log n)$ where $n$ is number of nodes in the heap. If all three operations could be done in $\mathrm o(\log n)$ time then we could sort $n$ arbitrary objects using this heap in $\mathrm o(n\log n)$ (just insert all objectes to the queue and $n$ times get maximum and erase it).