The Concise New Partridge Dictionary of Slang (2007) says:
burroo; brew; buro noun an unemployment exchange; the Department of Social Security. Form a Glasgow pronunciation of 'bureau' as in 'Employment Bureau. UK, 1937
On the brew means being unemployed or receiving unemployment benefits, and is similar to on the dole.
The oldest I found in Google Books for the exact phrase on the brew is in the New York Magazine (Vol. 2, No. 11) of 17 Mar 1969, in an interview with New York immigrant James Toner from Belfast, Northern Ireland:
"What do the Catholics do for work?"
"Go on the brew. You know the stayet. The Relief. Then they hang around the bars and the bookie shops. There's nothin' else they can do."
I've never heard it before, but I've never lived in Scotland. It is still used by people in the Glasgow area. Looking for current usage, I searched Twitter and found this from someone in Glasgow:
some c*ltic fans on this clearly should have been lawyers insted of sittin on the brew shut up and accept 54 titles and still goin strong
And this from someone in Cumnock (39 miles from Glasgow):
Might be my last week of work need a new job so am no back on the brew #badtimes
And finally, this from someone in Wishaw (15 miles from Glasgow):
Its a nightmare knowing that your monthly wage is going to be less than what a person on the brew gets in 2 weeks. This nation is backwards!