The British Library is without a doubt one of the finest libraries in the world. The library's holdings on Burma are particularly strong. There are multiple ways to search for materials, including Archives & Manuscripts, Digitized Manuscripts, Sounds (which include recorded interviews with British subjects on their first-hand experiences in Burma before independence), and Endangered Archives, featuring Burmese, Pao, Karen, Shan, and Mon palm leaf manuscripts. Also see the very useful collection guide to the library's Burmese collections.
The British Museum houses an enormous collection of unique artifacts from Myanmar, thousands of which have been digitized. Interestingly enough, the keyword search term "Myanmar" produces less than 100 items, whereas the term "Burma" retrieves nearly 4,000.
Launched in 2013 at the University of Toronto in collaboration with several other institutions and faculties, the Myanmar Manuscript Digital Library represents one of the most ambitious Burmese manuscripts digitization projects ever conceived.
The British National Archives in Kew has digitized well over 40,000 archival records of relevance to Myanmar/Burma, and thousands more that may be requested. In order to have full access to these collections, researchers are required to register for free. The Archives discovery tool is the easiest way to search all available collections.
The Online Burma/Myanmar Library is the largest open access repository and portal for articles, periodicals, reports, books and videos (and links which lead to many more thousands of documents). Topics include health, land, economy, environment, law, foreign relations and history. The site offers access to more than 60,000 documents.
The Digital Bodelian houses a small but unique collection of Burmese manuscripts and watercolor paintings depicting life in 19th century Burma.