Can faculty do research during a labor strike?
Consider the following FAQ from West Chester University discussing the possibility of a strike:


  
  Will the Pennsylvania State System of Higher Education universities, includingWest Chester University close if a strike is called?
  No. The universities will remain open if a faculty strike is called. By law, individual faculty members have the right to decline to participate in a strike called by the union so individual classes could still be held.
  


and at University of Northern British Columbia:


  I am doing research as part of my degree. Will I be able to continue during a strike?
  It depends. Some research may be allowed under the terms negotiated at the start of job action. Also, anyone may cross a picket line during a strike unless the Administration declares a lockout (in which case nobody is allowed in). You should discuss this with your supervisor.
  
  How will a strike impact the delivery of supplies, reagents, enzymes,
  etc.? During the 72 hours prior to the start of job action, the
  parties negotiate “essential services,” that is those things that are
  deemed essential. In general, anything not declared an essential
  service would be impacted, which would probably include most
  deliveries. You should discuss this with your supervisor.


And a report about the University of Puerto Rico (admittedly, this is a student-led strike):


  Since the University of Puerto Rico strike began, faculty and students have only had intermittent access to its labs. Even when they are allowed to enter, they are working with dwindling supplies of some chemicals and other research materials. Closed campuses mean new shipments cannot be delivered.


These suggest, in a formal capacity, some research might still be able to take place, either from crossing the picket line, or from negotiations for research that is time sensitive (for example, many biology experiments which must be monitored, changed at certain intervals, etc.) but that they may certainly suffer during long strikes.

Beyond that, one imagines that it's impossible to faculty not to do "research" in the more expansive sense of the word, as occasionally that involves idly musing on a subject, and there's no reason to suspect that faculty would stop thinking about their life's work during a strike. Rather, it's far more likely to impact classes.