After a triumphant tour of Japan, then the United States and ending in Italy, the Girl with a Pearl Earring has returned home to the Mauritshuis royal picture gallery in The Hague. For ever. The museum, which reopened last month after two years' renovation work, will no longer allow Vermeer's masterpiece out. Officially the Mona Lisa of the North has been gated in order to please visitors to the Mauritshuis who only want to see that painting. Its fame has steadily increased since Tracy Chevalier published her novel in 1999 followed in 2004 by the film by Peter Webber starring Scarlett Johansson. Anyone wanting to see the portrait will have make the trip to the Dutch city.

Girl with a Pearl Earring thus joins the select band of art treasures that never see the outside world. Botticelli's Birth of Venus never leaves the Uffizi in Florence; Las Meninas by Velázquez stays put at the Prado in Madrid; Picasso's Guernica remains just down the road at the Reina Sofia museum; and his Demoiselles d'Avignon can only be seen at MoMA in New York.

Other sedentary art works include La Joie de Vivre by Matisse, Le Facteur Roulin by Van Gogh and Les Joueurs de Cartes by Cézanne, which are unlikely ever to leave the Barnes Foundation in Philadelphia. It is impossible for the Isenheim altarpiece to leave the Unterlinden museum in Colmar, or for Degas' Petite Danseuse de Quatorze Ans to escape from the National Gallery of Art in Washington DC. Needless to say, the Mona Lisa is under lock and key in the Louvre, Paris.

So why are all these famous pieces so stay-at-home? Predictably the principal reason is their state of health. Many of them, including the Mona Lisa, were painted on wood and are very sensitive to climatic changes, making travel a major worry. This is equally true of anything made of wax, as is the case with Degas' original Danseuse, which is dressed in a silk bodice, a tutu and slippers, with a wig of real hair. "We cannot even imagine a situation in which we might loan it," says Deborah Ziska, head of public information at the National Gallery of Art. The versions seen elsewhere – in the Musee d'Orsay or the Metropolitan, for example – are bronze casts made in 1922, five years after the artist's death.

Many 19th-century paintings are fragile. "They were the victims of artists testing new materials which aged badly," says Sébastien Allard, head of the painting department at the Louvre. For instance, when Géricault painted Le Radeau de la Méduse, he used Judea bitumen as a primer. It took so long to dry that tiny cracks have formed on the surface.

As it happens, just the size of the painting (4.9 metres by 7.2 metres) makes it impossible to remove from the museum. The canvas would have to be rolled up, a practice most curators now consider far too risky. Many paintings in the Louvre never move, due to their dimensions. "When we wanted to restore Veronese's Wedding Feast at Cana (6.8 metres by 9.9 metres) or La Bataille d'Eylau by Gros (5.2m by 7.8m) we put up scaffolding and screens, and the restorers worked on the spot," Allard explains. "In the early 1990s, to move David's Sacre de Napoléon (6.8 metres by 9.8 metres) from the Mollien room to the Daru room, we had to cut notches in the door frames; otherwise it wouldn't have gone through." Loaning these king-size works to other museums is obviously unthinkable.

Crowds view Pablo Picasso's Guernica at the Reina Sofia museum in Madrid, Spain. Photograph: Denis Doyle/AP

Weight is a problem too. The job of restoring the Winged Victory of Samothrace is a case in point. The statue alone weighs 32 tonnes, not counting the 23 blocks of marble that make up the pedestal. There is no question of transporting it to the laboratory of the Centre for Research and Restoration of the Museums of France, at the other end of the Louvre. After 10 years' hesitation and preparation, it has finally been moved almost next door to undergo restoration.

But the last wishes of donors represent the most pressing constraint. When they leave their treasures to a museum many collectors impose strict conditions. The Van Gogh paintings at Musée d'Orsay donated by Paul Gachet's son can only be lent to another institution for retrospectives. Le Déjeuner sur l'herbe by Manet and Les Coquelicots by Monet , which originally belonged to Etienne Moreau-Nélaton, cannot be shown anywhere else.

"The same rule applies to La Jeune Fille au Jardin, by Mary Cassatt, for which we receive many applications," says Xavier Rey, head of collections at the Musée d'Orsay. "We always say no because the painting, along with our finest pointillist Pissarros, belongs to the group of works bequeathed by Antonin Personnaz, with no scope for loans. Obviously we always comply with the conditions for a donation, if only as a matter of respect, not to mention legality. But also not to put off future donors." This is an important consideration, because as all curators know well, museum collections would not be what they are without the generosity of private collectors.

The most extreme restrictions often concern museums set up by private collectors. Convinced that their collection is a work in its own right, they confine it forever behind the walls of an institution. Witness the Wallace Collection in London (one of the finest sets of 14th- to 19th-century paintings, including Fragonard's delightful Swing). The same applies to two-thirds of the Frick Collection in New York (the fortunate owner of three of the 37 known Vermeers). Albert Barnes, a successful pharmacist from Philadelphia, forbade his foundation from lending anything or even making minor changes to the disposition of his 2,500 works, which include 150 by Renoir, 69 by Cézanne, 60 by Matisse, several dozen by Picasso, not to mention some superb pieces by Monet, Toulouse-Lautrec and Van Gogh ... and the world's most beautiful Seurat, Les Poseuses.

The Duc d'Aumale, the fifth son of King Louis-Philippe, gave his estate at Chantilly to the Institut de France, along with his collection, which included three paintings by Raphael, three by Fra Angelico, five by Ingres and more works by Clouet than even the Louvre. They draw art enthusiasts from around the world. But this may not be good news for the Musée Condé. "The world of museums is highly competitive. It's a real handicap not to be able to loan works," says Nicole Garnier, the head of the museum. "It's quite simple, if you don't lend to others they won't lend to you." Which means it is difficult to stage exhibitions that attract the general public. Curators must work very hard to organise events that will draw local visitors, and convince them to repeat their visit, one of the key challenges for museums today.

"In scientific terms it really is a pity not to be able to take part in major international shows," Garnier adds. "That's where art history is happening now, through the confrontation of works." A well-designed exhibition does not simply bring together paintings or sculptures; it should also be an opportunity for curators to confront and examine works close-up, some of which were created centuries apart. Or to gauge the influence of one artist over another. "We [have sent] The Burial of Casagemas to the Prado for an exhibition on El Greco, who was a major inspiration for Picasso," says Sophie Krebs, head of collections at the Musée d'Art Moderne in Paris. "The museum in Madrid has nothing to lend us in return, but it's a very useful contact."

In 2013 Titian's Venus of Urbino was moved from the Uffizi in Florence to Venice after the museum's curator yielded to a request by then Italian premier Mario Monti. Photograph: Nicola Lorusso/Alinari Archives/Corbis

Some masterpieces are not quite as immovable as one would be led to believe. There is growing pressure for them to venture outside their safe havens, the number of exhibitions having steadily increased since the second world war and rising steeply since the 1980s. Not only are museums in need of funds, but they also have an image to promote. For example Daniel Percheron, the leader of the Nord-Pas de Calais regional council, is determined that the Louvre in Lens should one day exhibit the Mona Lisa.

Even the biggest attractions have to move sometimes. During the second world war the immovable Winged Victory was taken to the Château de Valencay, near Châteauroux, and the lady with the inscrutable smile was hidden under a curator's bed. Museums sometimes have to close for repair work, and rather than putting paintings in storage curators rent them out, to pay for part of the work. So the terms of Barnes's will did not prevent part of the collection going on a rewarding tour in the 1990s.

"A curator is by definition a rather schizophrenic character," Rey asserts. "It is their duty to protect their charge – which taken to its logical extreme means putting everything in the freezer – but also to show it to the largest possible audience."

"It's all a question of balancing risks and benefits," says curator David Cueco. "For example Delaunay's Equipe de Cardiff wasn't in very good shape when Suzanne Pagé, head of the Paris Musée d'Art Moderne, agreed to lend it to the MoMA for the High and Low show. But her move opened the way for some great Rothko paintings we'd never seen in France."

Lastly, curators are not immune to political pressure. In 2013, Antonio Natali, head of the Uffizi Gallery, allowed Titian's Venus of Urbino to be shown in Venice, yielding to the Italian premier, Mario Monti, in person. It was a symbolically powerful moment, the aim being to compare the old master with Manet's Olympia, the painting it had inspired, loaned by the Orsay. Even the Mona Lisa has taken part in two diplomatic missions, to New York in 1963 and Tokyo in 1974, stopping off on her way back in Moscow. Each time it was due to a request by the French president, against the curators' recommendations.

Would that be possible now? "I doubt it," says Allard. "Above all for its own protection. The context has changed too. There are far more visitors now: 9.3 million a year, compared with a few hundred thousand in those days. And over two-thirds are foreign visitors often on a once-in-a-lifetime outing to the Louvre ... to see the Mona Lisa." It would be terrible to disappoint them, just as for Girl with a Pearl Earring in The Hague.

"Museums are increasingly inclined to turn their masterpieces into icons, and in so doing into tools for communication," says arts management consultant Jean-Michel Tobelem. "Successfully so, what's more. The opposite extreme is the Pompidou Centre, which is packed with fabulous art none of which has come to symbolise the museum. Which is probably why there's rarely a big queue for the permanent collection. It's hard to strike a balance between investing excessive importance in specific pieces and the need for fame." To be on the safe side it is probably best to book ahead for the Mauritshuis.

This article appeared in Guardian Weekly, which incorporates material from Le Monde. The opening paragraph has been amended as it incorrectly stated that The Hague is the Dutch capital.