API to get citations from DOI
An increasing number of publishers are making their citations open. When this happens, you can get them from the Crossref REST API. 

This query lists all of the DOIs for which you can get the references, currently 20,455,807

http://api.crossref.org/works?filter=reference-visibility:open,has-references:true

Here's an example lookup. Look in the "reference" field:

http://api.crossref.org/works/10.1179/1942787514y.0000000039

You can also look at the Initiative for Open Citations and OpenCitations.net.

(I work for Crossref)