Lower bounds and $P$ vs $NP$
Every $NP$-complete problem is an example of this! It's just a question of how good the lower bounds are.

For example, there's an $\Omega(n)$ lower bound on basically every problem, since you need to read your entire input to solve it. This is a very imprecise lower bound, but it does exist.

So, you're not going to get a good list of a bunch of problems like this, because such a list would be too large, and uninformative. 

Whether there are problems with notable lower bounds is a different issue. I don't know of any that fit your criteria (sorting is the most famous one, but it's in $P$). Here's an example question about bounds on 3SAT:


https://cstheory.stackexchange.com/questions/93/what-are-the-best-current-lower-bounds-on-3sat


Not very informative, as you can see.
The known lower bounds in such a case will always be polynomial. If there were any problems verifiable in polynomial time (in $NP$) with an exponential lower bound, that would imply that $P \neq NP$.