Nobody loves a fallen dictator.

A week after fleeing tropical Haiti, Jean-Claude Duvalier has overstayed his extremely frigid welcome in France. He has made it clear that he would like to stay here permanently, but the Socialist government insists that it wants to get rid of him as swiftly as possible.

There is just one problem: No other country has been found to accept the man whose family ruled Haiti for nearly three decades.

When Duvalier, widely known as Baby Doc, arrived in France last Friday night, French officials said that he had been given permission to stay here for a week. They depicted the decision to admit him temporarily as part of a strategy worked out with the Reagan administration to avoid a bloody rebellion in Haiti.

For the past week, Duvalier has been closeted with members of his family in a luxury hotel in the lakeside resort of Talloires in southeastern France, living in a suite of rooms once occupied by former president Jimmy Carter.

French officials from President Francois Mitterrand down insisted today that there was no question of granting Duvalier's request for permanent political asylum. The former dictator's French lawyer, Sauveur Vaisse, has argued that his client could be a victim of "persecution" if he were sent back to Haiti.

In Washington, State Department spokesman Charles E. Redman said the United States is "working closely with the French" in their efforts to find a country to take Duvalier. He declined to elaborate.

Speaking informally to journalists, Mitterrand said today that the right of asylum was usually reserved for the victims of human rights abuses -- not the perpetrators.

French officials said they still had hopes that the West African state of Liberia might agree to accept Duvalier and his family. President Samuel Doe's government said earlier this week that it would be willing to grant Duvalier asylum but later backed down, saying that it would only consider an asylum request.

One official said that a possible solution might be to send Duvalier to the United States, the country that persuaded France to accept him in the first place, but conceded that this could only be done with Washington's agreement.

With little more than a month to go before crucial legislative elections, the Socialist government is anxious to avoid being dragged into an embarrassing political con-troversy that could arise if Duvalier's stay became permanent.

Snippets of news reaching the outside world from the heavily guarded Hotel de l'Abbaye, where Duvalier is staying, suggested that French President Mitterrand said the right of asylum was usually reserved for the victims of human rights abuses. the deposed dictator spends much of his time watching movies on video recorders that he has had installed in his suite. According to one of the hotel maids, he is also studying French driving laws, which evidently forms part of his passion for anything to do with fast cars.

Duvalier is also reported to have asked for the removal of five paintings from the top floor of the hotel on the ground that they irritated him. One of the pictures depicted a severe-looking judge while another was of the sacrifice of a cock.

A petition circulated by the mayor of Talloires demanding Duvalier's immediate expulsion already has been signed by 140 of the town's 600 residents. It has not, however, received much support from local traders and hotel-keepers whose businesses have received an unexpected fillip from the arrival of hundreds of journalists.

French officials pointed out that the government could easily find a legal justification for the expulsion of Duvalier, despite his pending request for asylum. The immigration law states that, in the event of a threat to public order, any foreigner can be deported without appeal.

The government has forbidden Duvalier from traveling to either of his two residences in France.

At least 15 members of the entourage that arrived with Duvalier a week ago already have left Talloires, leaving behind only his wife, Michele, and their immediate family, according to French sources.