In 91, a first class is not required unless the company requires it. Some companies require a first class once a year that can lapse to a second class after 6 months. Some companies add a requirement to take an even more thorough non-FAA physical at the 6 month interval after the FAA physical.
In 135, a first class is required if you operate in a position that requires use of an ATP rating, i.e. PIC of anything requiring a type rating. Once again, though, the company may have other policies over and above, especially if they participate in any accredidation audit programs, like Wyvern, etc.