All instruments with free metal reeds can, principally, be retuned. In the case of harmonicas, it is even sort of common practice.
There are quite a number of videos in the web showing step by step how retuning is done, for example by Håkan Ehn .
A lot of information on harmonicas and tuning can be found on Pat Missin's homepage.
Not to forget is that, in the case of the diatonic harmonica (aka "blues harp"), the playing technique known as "bending" already allows, within certain limits, infinite pitch variations, even without retuning. For example, it is possible to play a near-5edo scale on a standard harmonica (starting on E or "sort of Db" on a C harmonica):
Near-5edo scale, standard diatonic harmonica in C (amateurish attempt)
Besides, it is possible to play a CRP4 scale (starting on F, F# or G on a C harmonica) and a 4EDF scale (starting on F on a C harmonica) - both not up to the octave, but over the respective equivalence interval.
With a little training, licks in (subsets of) almost any tuning can be played. See the example in 15edo on the Porcupine Album Project page.
These points - and the fact that the instrument is small and cheap - make the harmonica a good candidate as a gateway to microtonal music.