The cheapest journals are typically those controlled directly by academics themselves, although they're not necessarily the most efficient.
There is now (2019) a more comprehensive study of the costs by Alexander Grossmann and Björn Brembs:
Here we provide a granular, step-by-step calculation of the costs associated with publishing primary research articles, from submission, through peer-review, to publication, indexing and archiving. We find that these costs range from less than US$200 per article in modern, large scale publishing platforms using post-publication peer-review, to about US$1,000 per article in prestigious journals with rejection rates exceeding 90%. The publication costs for a representative scholarly article today come to lie at around US$400. We discuss the additional non-publication items that make up the difference between publication costs and final price.
Another article from a while ago, Roger Clarke (2007), finds that for-profit publishers spend thousands of dollars per article on functions which a fully open access and non-profit journal doesn't need or want:
For–profit publishers have higher cost [...] much greater investment in branding, customer relationship management and content protection. [...] a computed per–article cost of US$3,400 compared with US$730 [for non-profit electronic journal].
However, prominent open science advocate Martin Paul Eve warns about The Problems of Unit Costs Per Article.
The problems are well illustrated by the very transparent article by eLife (2020), eLife Latest: The costs of publishing, which shows the differences between fixed and variable costs and the relationship with acceptance rate. Mind you, eLife is a very particular case because of its high selectivity and expensive ways of functioning, for instance it shells out hundreds of dollars to editors and reviewers (on average by published article, so less than half that for each reviewed article).
My own older summary follows.
Some universities run their own OJS instances, either in house or with some external contractors. Hosting often relies on existing infrastructure and staff time is often borrowed from employees the institution already has, so the costs are rarely easy to calculate, but we can figure out the order of magnitude.
For example, take the University of Bologna and the University of Milano: they publish 28 and 23 journals respectively (mostly in humanities), for a total of over 400 and over 600 articles per year respectively (according to DOAJ). For context, this size is comparable to top 15 publishers of OA Italian publications, where the biggest pure OA publisher has around 600 articles per year and the others vary between 500 and 2000.
They're both run with approximately 1 FTE "reserved" employee or less, as far as I know, which costs around 30 k€/year considering the national contract and pension contributions. Additionally they spend a few thousands euro/year on technical support. Even if you triple that amount to account for inefficiencies and unstated costs, that gives you less than 200 €/article in costs. Of course it's just an example for their case.
Some other publishers (typically public research entities or consortia) are transparent enough that we know their costs to provide certain services. See for instance: