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.
You might want to look into more modern clustering algorithms. Anything that runs in O(n log n) or better should work for you. There are plenty of good reasons to not use hierarchical clustering: usually it is rather sensitive to noise (i.e. it doesn't really know what to do with outliers) and the results are hard to interpret for large data sets (dendrograms are nice, but only for small data sets).