get from Boston airport to Burlington Marriott hotel by public transport?

At your terminal, you'll need to find out how to get to the Airport Station. Shuttle buses have been replaced by the Silver Line bus. On a weekday, take the Blue Line inbound train to the State Street station. Assuming there are no delays, this leg of the trip is less than 10 minutes. From State Street, you can take either the 352 Burlington Via I-93 bus or the 354 Burlington Via Woburn Sq & I-93. The 352 lets you off at 76 Cambridge Street, reportedly just a three minute walk from the hotel. The 354 lets you off at the intersection of Cambridge St And Mall Road, apparently directly at the hotel. The entire commute takes between 70 and 130 minutes depending on the timing between buses.

If you're arriving on the weekend, it becomes more complicated. Take the Blue Line inbound train to Government center. From there, you'll need to take a Green Line trolley (follow the signs or ask someone) to Park Street station. At Park, you'll need to take the the redline train to Alewife (it's the last stop going in that direction so make sure you aren't on the side for the Ashmont/Braintree trains-- the opposite direction--and you can't go wrong). From Alewife, you get the 350 Burlington bus, which also lets you off at 76 Cambridge.

Since bus times can vary wildly, it's a good idea to call the Transportation Authority's customer support ahead of time for the specific information on the day and time of your arrival: 617-222-3200 or 800-392-6100.