I don't know if this is the correct subreddit to ask this, but I remember a while ago I read on a thread here something about museums having copies of the original artwork on display, while the originals are kept safe on a deposit or something like that. Is this true? Is this a general sort of international rule or something that only a few museums do, only with some artworks maybe?
My museum has mostly original artwork and artifacts on display, but it is a old county museum that has historically not had museum professionals working at it. For me, as I am updating exhibits (sometimes for the first time in 50 years), if the artwork or artifact is very fragile, has a chance the public might be able to easily touch it, or has the possibility of getting to much sun exposure then I will pull it and replace with a similar object if not an exact copy (funds permitting, which is very rare). Thankfully, my museum has tinted windows and LED lights, so as long as the case is secure then I don't worry about the object.
When I worked for a larger state museum with a historic grounds, though, almost everything was being constantly rotated through exhibition. So an item may spend a few months on display and then sit in storage for up to ten years before it is put on exhibit again. The "copys" were used in areas where public interaction was expected, like the historic houses.
While this does not really deal with artwork, I hope it sheds some light on your question.
Many artworks are made with materials that degrade over time. For this reason it can be tricky to display objects that are very old, damaged, or at risk of decay if exposed to the public.
Trace amounts of acid in a paper or medium (paint etc) may not degrade an artwork within the artist's lifespan, but given enough time it will cause issues. Oxidation, when a work is exposed to air, can also cause problems -- this is why paper yellows with age, for example. High humidity obviously introduces moisture which is no good, but even worse is fluctuating humidity (when it rises and falls significantly) which can cause objects to warp and degrade.
Light, and especially sunlight, is harmful to the longevity of most things but is a particularly severe risk for organic artifacts, textiles, and works on paper like old books, drawings, prints, documents etc. If you see old documents on permanent exhibition it is probable that they are copies because being exposed to light 24/7 all year would fade them beyond repair. When displaying originals, museums often put these objects in rooms with dim lights and for short periods of time.
It's also important to remember that the majority of objects that museums collect have had long lives before their arrival, maybe several hundred or even several thousand years, which is a lot of time for mishandling, abuse, and the environment in general to eat away at the integrity and stability of an object. While art conservation today is pretty amazing, it can be difficult or impossible to repair some objects to the degree that they can be displayed without risking further damage.
I can't think of any objects off the top of my head that are replaced with copies for security purposes. The allure of seeing real art, in person, is what draws people to museums, even if they have to stand at a distance or look through protective glass.
To answer your question directly, conservation is perhaps the most important responsibility a museum has and so any respectable museum anywhere will do this for sensitive items.