Hi, The session at the Brendan Behan, in Jamaica Plain, takes place Saturdays from five to eight. Second-hand word says it's possible to sit in, and the Behan is a great small pub.
There is music every night at the Burren (dancing 8-10 on Modays), in Davis Square in Somerville, but good luck sitting in.
A session takes place every Sunday in South Eason, MA (about 1/2 hour south of Boston or north of Providence), on Rt 138 1/4 mile up from junction with 106. A lot of singing there, but on any given night, plenty of tunes too. The owner sits in to sing, and what is supposed to last from five to eight could run all night. Literally. Lots of big names in Irish and folk music have played the Blackthorne--suprising for a place out of the city. Some have sit in. In fact, the Blackthorne fits under the other thread for favorite pubs: very friendly, tons of beers, enthusiastic owner.
Some good sessions weeknights out of the city too, in Quincy. Hope this helps.