Do aircraft manufacturers like Airbus or Boeing charge the airline, when they assist in aircraft accident investigations?
Investigating agencies are in charge of most air accident investigations not involving criminal acts. In the US this is the NTSB, which is required to investigate air accidents per US Code Title 49, Chapter 11. These agencies exist to provide independent investigation of accidents and provide recommendations to regulators.

The investigating agents will do their best to understand the circumstances and causes of an accident. Agents of the NTSB have authority under §1134 of the above mentioned law to gather information about an accident.


  (a) Entry and Inspection.—An officer or employee of the National Transportation Safety Board—
  (1) on display of appropriate credentials and written notice of inspection authority, may enter property where a transportation accident has occurred or wreckage from the accident is located and do anything necessary to conduct an investigation; and
  (2) during reasonable hours, may inspect any record, process, control, or facility related to an accident investigation under this chapter.


Investigations can also benefit from working with people that are already familiar with the critical elements. As part of the investigation, the agency may invite various parties. This could include airlines, manufacturers, air traffic control, workers unions, other agencies, or anyone else with information or expertise that could be helpful. If these parties do not abide by terms for participation, this status can be revoked. The resources spent by parties during an investigation can in some cases be fairly intensive. For example, while supporting the investigation of the crash of British Airways Flight 38, Rolls-Royce and Boeing ran full-scale tests of the engine and the fuel system designs.

Investigating agencies are funded by their governments. The Aviation Safety department of the NTSB has an annual budget of about $50 million. Parties to the investigation represent their respective parties and are paid by their party to do so. Part of the party agreement acknowledges this:


  Participation as a party to an NTSB investigation is a privilege and confers no rights or benefits.


They "pay" that parties get is good press for assisting in the investigation, and helping to both prove and improve the safety of their products and services.