Under what circumstances can Part 91 allow compensation for air transportation?
The only way you can carry passengers for hire as a common carrier without an air carrier cert (either 121 or 135) is the sightseeing exemption -- 25 SM, landing at point of takeoff. However you will still need an LOA and to participate in an approved drug enforcement program.

If there is a 'commonality of purpose' e.g. all the passengers are employees of the company that owns the airplane, then it isn't common carriage, and you don't need the air carrier cert. This is how corporate flight depts operate, they're Part 91. (But they can only fly employees etc., they cannot fly the general public.)