Can a foreign national who has timely applied for a new work visa work without pay or "volunteer" for the company?
No. "Employment" for these purposes doesn't just mean getting paid; if the individual is performing productive work for the benefit of an employer, and taking direction from the employer, are generally considered to be employed. Generally, employees who perform productive work for the employer under the guidance of the employer - even if not for pay - are working without authorization and any employer who accepts such volunteer work without paying the employee is violating Federal labor law.
However, note that individuals on valid H-1B visas who are changing to a new employer may work for that new employer once the new H-1B petition has been filed, without waiting for approval. This is a special situation and normally this won't be a situation where the old-employer H-1B is expiring.
We desperately need this employee to begin a project and have timely filed an application for the appropriate visa. Is there any way to speed up the p
One possibility, if not already done at the time the petition is filed, is to pay U.S. Citizenship and Immigration Services the extra $1,000 for premium processing if the case type permits it (premium processing is available only or certain visas). This ensures 15-day turnaround from USCIS (not necessarily approval - the response may be a Request for Evidence or Denial as well).
There are occasionally other ways to permit the new employee to be working faster, but these options depend upon the company's structure and flexibility, the nature of the work, and other situation-specific factors. Consult an immigration lawyer for an examination of the circumstances and potential options.