How can I get a list of citations for a DoI or PubMed ID? I'd like to be able to submit a DoI or PubMed ID and find all the papers that cited that paper. 

In other words, I need a publicly-available citation graph database that covers papers in PubMed. 

The bottom comment on this question refers to a publicly-available citation graph: How do I get a citation dataset with full text articles in PDF format? but does not give any details of where it can be found. 

Does anyone know of any such citation databases?

UPDATE: To be clear, I really need a database - not a service like Google Scholar - because I want to do it programmatically for many hundreds of thousands of articles.
Quick and dirty version: Google Scholar

Type-in the DOI and read the citation count on the field "Cited by". Google Scholar does not filter for quality, so the citation count will include self-published papers, fake publications, etc. Expect some noise.

Better version: Web of Science

Ask your institution's librarian how you can access the Web of Science database and how to use it to retrieve the data that you need. That might require you to walk in daylight to the physical location of your library. Many survived that journey.