I doubt this is an issue to the vast majority of photographers coming to Mt. Rainier NP, but each photographer who's focus of their visit is photography should understand the rules for photographing in NP and if there is a need to get a permit. I will summarize the informtion here, but you can get a complete information from the NPS Website.
First, as stated by the NPS:
"It is the policy of the National Park Service (NPS) to allow filming and photography when it is consistent with the protection and public enjoyment of park resources and does not interfere with the public’s normal use and enjoyment of the park. Permits are required if the filming, videotaping, sound recording, or still photography:
• Involves the use of a model (or any on-camera talent), set, or prop,
• Involves taking photographs of vehicles or other articles of commerce for the purpose of commercial advertising,
• Could result in damage to park resources,
• Could result in significant disruption of normal visitor use,
• Requires access to areas normally closed to the visiting public.
Generally, permits are NOT required for:
•Visitors using cameras and/or recording devices for their own personal use,
•Sound technicians, and film or video news crews at breaking news events,
•NPS filming or photography, Department of the Interior Audiovisual Center filming, or filming/photography done pursuant to a cooperative agreement or contract."
This is common for almost all the NP's as an agency policy, exceptions where noted with the individual NP.
If you think you need a permit, it's best to call and talk with the NPS staff and consider getting one. This is important if you want to photograph in areas that are closed or restricted to the public or requires the interruption of visitors' experience, meaning you'll be in the way with your work or equipment.
In addition, larger groups of photographers, especially part of workshops, large photo tour groups or other situations, will have to apply for a permit through the annual process, see the Special Use Permits. These are separate from individual permits due to the larger group and possible disruption in the NP.
In the end, though, 99+% of photographers are free to pursue their photography. After that it's a matter of the photographer exercising common sense in their work not to damage anything in the NP or disrupt other visitors. Remember, you're one of many, and you don't want your experienced effected by someone else.