To elaborate on the title: Among somatic, post-mitotic cells, would the same intron on a given chromosome have the same sequence among all cells descended from a progenitor cell?
I believe that the kind of mutation that you are interested in is somatic mutation, as opposed to germ-line mutation.
Any given genomic locus (intron, exon, intergenic region) probably doesn't mutate most of the time. In the case of somatic mutation, it is yet more likely that no mutation happens.
Nonetheless, such mutations do happen, in all of those different regions. For example, here is a study that catalogs somatic mutations in introns that lead to a particular phenomenon called intronic polyadenylation; here is another that catalogs intronic mutations that mess up splicing.
The same is true for protein-coding somatic mutations, one can find similar catalogs of mutations for those.
Such mutations in different regions (intronic, exonic, and otherwise) are all of interest as events that can lead to cancer.
None of this should distract us from that fact that usually mutations don't happen at any given site. But it's a question of probabilities, not of absolutes.