Classic hierarchical clustering approaches are O(n^3) in runtime and O(n^2) in memory complexity. So yes, they scale incredibly bad to large data sets. Obviously, anything that requires materialization of the distance matrix is in O(n^2) or worse.

Note that there are some specializations of hierarchical clustering such as SLINK and CLINK that run in O(n^2), and depending on the implementation may also only need O(n) memory.