From the folks manning the Twitter and Universal Hub Aquapocalypse desks:
Back Bay: Dunkin' Donuts in the Prudential Center; Finagle-a-Bagel, 536 Boylston St.
Beacon Hill: Capitol Coffee House, Bowdoin Street, next to the State House.
Cambridge: Pretty much every single place that normally serves coffee, because they're special.
Chinatown: Au Bon Pain at New England Medical Center.
Dorchester: Sugar Bowl Cafe on Dot. Ave. near JFK/UMass (cash only); Flat Black Coffee has iced coffee available at Dorchester Lower Mills and Ashmont stores.
Downtown: Mulligans on Canal Street; Cosi at South Station (trucked in from Cambridge); Plaza Deli, second floor, 2 Center Plaza; City Hall Deli, eighth floor, Boston City Hall; Max's on Milk Street; the Milk Street Cafe; Flat Black Coffee on Broad Street has hot and iced espresso drinks; Finagle-a-Bagel in Government Center; Sip Cafe in Post Office Square Park; Grassroots Cafe, 101 Arch St.
Hyde Park: TC's, Fairmount Avenue.
Jamaica Plain: Dunkin' Donuts, Centre Street (across from the fire station).
Longwood Medical Area: All the hospitals have their own water supply, so everything should be running. Au Naturals at Brigham and Women's for example.
Milton: Dunkin' Donuts, East Milton Square.
Boston Common on Salem Street. All out, but will have more Tuesday morning.
Roslindale: Select Cafe, Roslindale Square (across from the train station).
Roxbury: Peet's at Northeastern International Village, 1175 Tremont. No espresso, though.
Somerville: Mr. Crepe in Davis Square; Blue Shirt Cafe, Davis Square; Aroma Cafe (formerly Cafe Rossini and Cafe Lola), on Highland Ave. just past Cedar St.; The Biscuit, 406 Washington St.
South Boston: Doughboy Donuts, 220 Dorchester Ave.; Flour by the Children's Museum.
South End:: Dunkin' Donuts at Mass. Ave. and Columbus Ave.; Mohr & McPherson Cafe, 460 Harrison Ave.; Mike's City Diner on Washington Street (closes at 3).
Watertown: Joe's Variety, Orchard Street.
Across the area: Au Bon Pain shops have coffee; except the one at the Pru.