Distributions which are intractable to sample from?
I'll expand my comment to an answer. Many combinatorial structures in graphs are actually NP-hard to sample from.

The earliest example I can think of is JVV86 (Thm 5.1), which shows that there is no polynomial-time algorithm to randomly generate a cycle in a directed graph, unless NP=RP.

Similar results along this line are NP-hardness for sampling independent sets, colourings etc. See e.g. DFJ02, Sly10, GSV15. Note that the corresponding search problems are all easy. 

The problem can also be defined by "soft" constraints. E.g., sampling from an anti-ferromagnetic Ising model is NP-hard. See JS93 (Thm 14).

Results of this sort are usually stated as NP-hard approximate counting problems, due to the standard self-reductions JVV86.