Does the NTSB and other investigative agencies charge airlines for investigations?
In the United States, no, the airline does not pay the NTSB or the FAA for accident investigations. This is considered a public service and is paid for by tax funds. 

That doesn't necessarily mean that they won't pay for anything. For example if the NTSB has to remove wreckage to examine it, they will often have the insurance company pay for the removal and transport. This is especially true in GA investigations where the responsibility for moving the wreckage (after the NTSB says so) is up to the owner/their insurance.

Whether or not Boeing/Airbus/XXX charges airlines would be another matter that is probably outlined in contracts between the airlines, the aircraft leasing company, and the manufacturers. 

Note that NTSB may involve any number of parties, for example they would have Boeing for the airframe, GE for the engines, Honeywell for the avionics/sensors, and representatives from other companies/groups (like the Airline Pilots Association) as they see fit. NTSB has manufacturer representatives for the bigger parts, and may ask for additional help from other smaller providers. This is done under the NTSB "party system" which is by invitation and does not compensate the parties for participation.

So as far as accident investigation goes, no, airlines/manufacturers don't pay for that directly.