Ah...better read 91.501(b) a little more closely.
Some of the highlights...
(b) Operations that may be conducted under the rules in this subpart instead of those in parts 121, 129, 135, and 137 of this chapter when common carriage is not involved, include -
(5) Carriage of officials, employees, guests, and property of a company on an airplane operated by that company, or the parent or a subsidiary of the company or a subsidiary of the parent, when the carriage is within the scope of, and incidental to, the business of the company (other than transportation by air) and no charge, assessment or fee is made for the carriage in excess of the cost of owning, operating, and maintaining the airplane, except that no charge of any kind may be made for the carriage of a guest of a company, when the carriage is not within the scope of, and incidental to, the business of that company;
(8) The carriage on an airplane of an athletic team, sports group, choral group, or similar group having a common purpose or objective when there is no charge, assessment, or fee of any kind made by any person for that carriage; and
Skip down to Part D of that paragraph, and read:
(d) The following may be charged, as expenses of a specific flight, for transportation as authorized by paragraphs (b)(3) and (7) and (c)(1) of this section:
(1) Fuel, oil, lubricants, and other additives.
(2) Travel expenses of the crew, including food, lodging, and ground transportation.
(3) Hangar and tie-down costs away from the aircraft's base of operation.
(4) Insurance obtained for the specific flight.
(5) Landing fees, airport taxes, and similar assessments.
(6) Customs, foreign permit, and similar fees directly related to the flight.
(7) In flight food and beverages.
(8) Passenger ground transportation.
(9) Flight planning and weather contract services.
(10) An additional charge equal to 100 percent of the expenses listed in paragraph (d)(1) of this section.
Even those charges are only applicable to demonstration flights to prospective customers (when no charge is made, except those charges), and the carriage of property in furtherance of a business (again, when no charge is made except those charges outlined above).
The question here is w(h)eather a private operator may auction off time in a jet, for charity. The person donating the money to charity pays the charity, and the owner doesn't take the money. The operator then attempts to claim that as he or she received no benifits from the flight, then no charge was made, and therefore the flight is legal.
However, the customer has paid for the privilege of making the flight, regardless of the entity to whom the money was paid.
Not enough information has been provided to answer the question. A key issue is where the flight is going. Is it returning to the same point of departure, or is it making transit between two points, and thus transporting persons or property from one point to another? Sounds dangerously like an illegal part 135 operation, if that's the case.