I had a lot of misconceptions before researching this. I assumed many leaves would become so brittle, they would crumble at a touch, and specimens would slowly degrade, due to bacteria and mold, and chemical reactions within the plant material and with air and humidity. I envisioned needing to initially disinfect the specimens, then after drying, hermetically sealing them away from air, humidity, and handling. It turns out the specimens are actually pretty tough and stable.
Bacteria and fungii don't feed on the specimens once they're dry. There can be some color changes during the drying process, but dehydration stops the kind of chemical activity that goes on inside the plant. Mounting the specimens minimizes flexing and adds a little strength.
As far as environmental conditions, keeping the ambient temperature and humidity stable, and the humidity below the point that supports mold growth, is apparently enough to keep the specimens in good shape. It isn't necessary for them to be hermetically sealed.
Collections at museums and universities can date back hundreds of years. The storage rooms are probably controlled for temperature and humidity, but the specimens are just mounted on heavy, acid-free paper, and basically stacked. A typical setup is cabinets containing stacks of what look like acid-free manilla folders. Within the folders are bundles of specimen sheets with a protective sheet between them. Specimens that are too thick to mount on paper are just stored in boxes.
So it looks like protecting the specimens is mostly at the front end. Once they're properly pressed, dried, and mounted, they don't need more than common sense precautions in storage and handling.