You need to look to your city's landlord/tenant ordinances to see when you may turn off the heat and, conversely, when you must turn it back on. Got to Burlington's website and see if it lists the person you should talk to - probably the building code administrator would be a good place to start: http://www.burlington.org/
There are laws and regulations that address what a landlord must provide. Heat at certain temperatures must be available to a tenant during specific times of the year from about mid-September to mid-June. While a landlord may be entitle to turn off the heat at certain times of the year, it would be prudent to speak to an attorney about ones specific circumstances, rights and remedies. A dispute can be costly, even if needed to just prove you are correct.
There may mechanical options to address the situation, such as thermostats that will not allow the heat to be increased above the legal requirements. How the units heat is controlled is important. If the rental has a separate furnace, it should be investigated as to whether the tenant can be responsible for the cost to heat.
The cost of speaking to an attorney will be well spent so you are fully advised about your rights as they relate to this specific situation.
You can find some general information about landlord/tenant law here: http://www.lawlib.state.ma.us/landlord.html
This answer is provided in response to a "hypothetical" question and provided for general, informational purposes and does not create an attorney/client relationship. The information presented is not legal advice and may change based additional information and research. It is recommended that you speak to an attorney to discuss your specific legal issues.