Wikidata has a lot of data about embassies and consulates.
I created a Python+SPARQL script that pulls that data and maintains an easy-to-consume CSV file. You can download the CSV file at https://database-of-embassies.github.io.
Data sample:
operator;operatorQID;jurisdictions;jurisdictionQIDs;country;countryQID;city;cityQID;address;latitude;longitude;phone;email;website;facebook;twitter;youtube;picture;pictureAuthor;pictureLicense;pictureLicenseURL;type;typeQID;creation;QID
Afghanistan;http://www.wikidata.org/entity/Q889;Austria;http://www.wikidata.org/entity/Q40;Austria;http://www.wikidata.org/entity/Q40;Vienna;http://www.wikidata.org/entity/Q1741;;48.2018;16.3738;;;http://www.afghanistan-vienna.org;;;;;;;;embassy;http://www.wikidata.org/entity/Q3917681;;http://www.wikidata.org/entity/Q104602592
France;http://www.wikidata.org/entity/Q142;United States of America;http://www.wikidata.org/entity/Q30;United States of America;http://www.wikidata.org/entity/Q30;Washington, D.C.;http://www.wikidata.org/entity/Q61;;38.9137;-77.0779;+1-202-944-6000;;http://www.ambafrance-us.org/;FranceInTheUs;franceintheus;UCbq_AyHPdsFDCFgGrjSeMKQ;http://commons.wikimedia.org/wiki/Special:FilePath/France%2C%20Washington.JPG;;CC-BY-SA-3.0;http://creativecommons.org/licenses/by-sa/3.0/;embassy;http://www.wikidata.org/entity/Q3917681;;http://www.wikidata.org/entity/Q2841718
The GitHub page also lists things that you can easily do to help improve the data.
I would say the Wikidata data is more structured than OpenStreetMap data, for instance consulates are clearly marked as such. The CSV file currently contains 6365 diplomatic representations, though not all with coordinates.
License: Public domain
A nice plus is the pictures, which can for instance be used as thumbnails:
The URL, license, author of each embassy/consulate's picture are also present as columns in the CSV file.