Few papers use DOIs in the reference section (example below). What are the downsides of using DOIs when listing references?
A typical example from one of the main conferences in the natural language processing field showing no DOI:
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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").
There's no downside for me, but finding them can be challenging. I just looked at several recent downloaded articles, and I don't see a DOI on the front page. So, I'd say my personal inertia keeps me from digging around on the Internet looking for a DOI to add to my BibTeX entries for each paper I want to cite. I try to add them when they're immediately presented on the papers I cite, since they do have to be found. But having to go look to find them makes them less likely for me to use. Not every paper has a DOI that is readily found. There's no step in my workflow when writing where I go through all my DOI-less citations and try to find one. Maybe that's a bad habit, but it is my habit. I suspect that's true for many authors, but I think you have a bad assumption that there's some downside to DOIs that prevents people from using them. It's not enough of a requirement yet that people do it because journals or the community demand having them on every citation so that people get over their inertia and go find them or figure out if every citation even has one.
I can speak for mathematics, where virtually no papers are cited with DOIs. Indeed, I confess to only a vague understanding of what they are (and I suspect most mathematicians would say the same).
The only downsides of which I'm aware are that they would take some effort to look up, and take a little bit of space. I think the reason they aren't used is because mathematicians don't see any obvious need or benefit. At any rate, I glanced through the DOI FAQ just now and didn't see any compelling use case.
Perhaps this is because we have MathSciNet (Mathematical Reviews), a searchable database of essentially every modern math paper that has been published in a reputable journal, with reviews and links to online versions (when available). Indeed, sometimes citations include the "review number" from this database. I don't bother, because it is just as easy to look up papers by author name and title, but some people do.