Is there any API service to provide just the number of citations by DOI of an article?
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
Here's an example lookup. Look in the "reference" field:
(I work for Crossref)
If you are a publisher member of CrossRef, you could use their Cited By service, and enumerate the results from that. If you had access to that, though, I expect you wouldn't be asking your question here.
You could try their REST API, although I don't know what information you can retrieve via that route.
The Scopus API (Elsevier) looks promising, and there seems to be a free version.
If you have access to Web of Knowledge/Web of Science, they also have an API.
SciFinder also has an API.