I don’t think there is. The only way I can think of is to compare the journal to a list of open access journals. It’s possible to get the journal from the DOI using Crossref’s content negotiation (http://www.crosscite.org/cn/).
That won’t be perfect though. You will get false negatives, e.g. for papers that are open access in a closed access journal (because the authors paid or some other reasons). Also it’s unclear if a closed access paper after the embargo time ican be called “open access”.
There also will be false positives for example for journals that switched recently to open access and have still old papers that are closed access (@andreas mentioned Nature Communications as an example).
If you are just interested if people can get the paper or not, the only way is to try and see if you can download it or hit a paywall. That’s highly non-trivial because going from the DOI to a PDF download link is full of problems. We have written a 85% of a blog post about it already which we hope to release in the coming weeks.
Don’t expect too much. Actually the blog is more about the problem not so much about the solution which requires to have full-time staff to have a remote chance to maintain an up-to-date automatic PDF downloader.
For Hybrid Journals we need to look to publishers and Crossref to make metadata available on the “openness” if a publication using the [NISO standard][1] (i.e. open or not, and link to reuse licence). But that just covers ‘gold’ OA. For ‘green’ OA papers, Research Councils UK is asking UK Research Organisations to implement the [RIOXX Application Profile][2] in their Institutional Repositories, allowing harvesting of information on embargo periods, funders and GrantIDs.
“DOAI (Digital Open Access Identifier) is an alternate DOI (Digital Object Identifier) resolver that takes you to a free version of the requested article, when available.”.
I wonder if Paperpile could search it also when looking for a PDF (as it does with dx.doi?)
If there is an open access paper available Paperpile usually does not have problems getting it. We also have some fallbacks (e.g. we get the PubMed Central manuscript if the journal version is restricted). If we don’t get an open access paper that’s considered to be a bug and we need to fix it.
The real challenge however is to get the non-free papers using all sorts of paywalls, proxies, subscriptions etc. So I don’t think doai.io will add much to our PDF importer performance.