What are the downsides of using DOIs in reference sections of publications?
To me, the primary reason for not including DOIs in the bibliography is the length restriction of the paper1. The DOI on its own doesn't work as a bibliographical entry, together with the human-readable information, it is redundant, and it adds roughly one line to the paper per entry. Therefore, a bibliography with DOIs is (by removing the DOIs) a welcome source of extra space for content that would otherwise have no space left.

1: In my field, length restrictions are typically tight (8 to 12 pages in two-column formats for "full papers") and to be taken seriously, and additionally aggravated by the fact that papers in my field typically include plenty of figures. Moreover, in my field, bibliographies are normally included in the length restriction or subject to their own length restriction (I have come across rules like "up to 8 pages, or up to 9 pages, if the last page only contains bibliography items").