I usually associate the standard deviation with the mean and the IQR with the median. Is there a measure of dispersion typically associated with the mode?
There isn't really a typical measure of spread associated with the mode.
Of course one could calculate a root-mean-square-deviation-around-the-mode (a standard-deviation measure with the mode taken as the center) or a mean-absolute-deviation-around-the mode, but neither of those are common measures.
A measure of spread that is sometimes used with the mode is the width of the smallest interval containing half the data (the shortest half, or shorth). This measure is discussed by Nick Cox here, but it isn't "about the mode" -- the shortest interval containing half the data needn't include the mode (indeed there may be multiple such intervals, and any or all of them may fail to include the mode).
However with a continuous unimodal distribution, if mode and the shortest half are unique, then the shortest half will contain the mode (otherwise, some shortest half will contain a mode).
[Incidentally that linked article on unimodality does mention a result relating to the root mean square deviation around the mode, indicating that the first measure I mentioned does crop up in some situations.]