This is one of those words in which a part comes to represent the whole (there’s a term for such words which is on the tip of my tongue but like to remain there for the time being.) It originally meant the baize material that covered a table for accounting, then came to mean the table itself and eventually the room in which such a table stood, then the office or building. Here’s OED.
Etymology: < French bureau baize which covers a table used for accounting (1316 in Old French as burel ), table used for accounting (1361 in Middle French), table at which audiences are granted (1477), room in which audiences are granted (containing a table for this purpose) (a1483), accounting office (1495), writing desk (second half of the 16th cent.), office used for the transaction of government or public business (1557; 1680 in extended use, denoting any kind of office), specific semantic developments of Old French buriau , Old French, Middle French burel , Middle French bureau coarse woollen stuff, baize (used for covering writing desks) (see burel n.1).
The stress was once on the final syllable in the UK, apparently shifting to the first from about the 50s onwards, although in the US the stress had been on the first since at least the early 19th century.
The stress appears to have originally been on the final syllable. N.E.D. (1888) notes this as the usual position of stress in British usage, and British 19th-cent. and early 20th-cent. pronouncing dictionaries record final-syllable stress as the only or chief variant ( D. Jones Eng. Pronouncing Dict. gives first-syllable stress as the preferred pronunciation from ed. 11 (1956) onwards). By contrast, 19th-cent. and 20th-cent. U.S. dictionaries, beginning with Webster (1828), record only the form with first-syllable stress.