What will air traffic controllers do when radio contact is lost?
You didn't mention any specific jurisdiction but for the FAA, the expected actions are in the ATC orders. Section 5-2-8 says:


  When you observe a  Code 7600  display, apply the procedures in para
  10−4− 4, Communications Failure


Section 10-4-4 gives general instructions for comms failure, it's too long to quote in full but the first paragraph summarizes it:


  In the event of lost communications with an aircraft under your
  control jurisdiction use all appropriate means available to
  reestablish communications  with the aircraft. These may include, but
  not be limited to, emergency frequencies, NAVAIDs that are equipped
  with voice  capability, FSS,  Aeronautical Radio Incorporated (ARINC),
  etc.


The instructions suggest some specific actions:


Broadcast via NAVAID frequencies
Ask the aircraft to use its transponder or make turns to acknowledge instructions


If comms haven't been reestablished within 5 minutes, ATC is instructed to consider it as suspicious:


  If radio communications have not been (re) established  with the
  aircraft after 5 minutes, consider the aircraft’s or pilot’s activity
  to be suspicious and report it to the FLM/CIC per FAA Order JO 7610.4,
  Chapter 7, Hijacked/Suspicious Aircraft Reporting and Procedures


Section 10-2-5 also says that lost comms are an emergency from the ATC perspective and should be escalated to the ARTCC. If comms aren't restored within 30 minutes then the aircraft is considered overdue, which triggers another set of procedures listed in section 10-3.

As for clearing airspace or whatever, there are no specific instructions in the orders, it just says that:


  When an IFR aircraft experiences two-way radio communications failure,
  air traffic control is based on anticipated pilot actions.


In other words, ATC expects IFR pilots to follow lost comms procedures as per 14 CFR 91.185 and will direct other traffic accordingly.

Finally, one 'routine' lost comms scenario is when ATC can't contact an aircraft because of poor radio coverage. This happens a lot with smaller aircraft (which fly relatively low) operating in remote areas and ATC might ask another aircraft, usually a higher-flying airliner if possible, to contact the first aircraft and relay their call: N12345, this is BigAir 123 with relay from Denver Center.