In the film Casablanca, there is a scene where a Nazi loudspeaker car (Gestapo according to Rick) arrives in Paris ahead of the German Army and tells civilians how to behave when the Army arrives. Is this accurate?

Yes, it is, albeit the loudspeaker cars were not Gestapo but rather the Wehrmacht itself after the advance guard that enter the city at 5:30 in the morning of June 14. The French had declared Paris an open city, meaning that no defense was offered on part of the French there in order to not reduce the city into ruins. With the advance guard approaching the center, German troops entered the city and had military vehicles with loudspeakers (most likely borrowed from the Wehrmacht Propaganda Companies) drive through the city instructing the citizens to stay in-doors.

The Guardian cites an Associated Press correspondent describing the city before the German arrival:

It was a strangely empty Paris that awaited the arrival of the Germans. The scene is described by a Press Association correspondent who left the city late on Thursday. Only a few police were to be seen in the streets. A handful of soldiers and some civilians wandered about aimlessly. Some housewives were doing their shopping at the only baker's shop still open in the West End. A mobile guard was posted at the door of the shop, a gun slung over his shoulder. There was no sound of gunfire or of exploding bombs.

The only authorities left in the capital were Cardinal Suhad, the Archbishop, officials of the essential services, and the Prefecture, Mobile Guards, and firemen. All bridges had been left intact, but the French are reported to have blown up the big armaments factories in the suburbs.

The main German forces entered the city at noon yesterday. They came from the north-west and by the Aubervilliers Gate from the north-east. From the north-western suburbs they marched through the west end down the Champs Elysées - tanks, armoured reconnaissance cars, anti-tank units, and motorised infantry. Machine-gun posts were set up at important points, and the wireless stations were seized.

The people left in Paris watched the entry in silence, reports the Associated Press correspondents. Small groups of people still sat along the terraces and boulevards and in the cafés. Shops were boarded up. In the Place de l'Opéra stood a solitary motor-car with a big "for sale" sign. The Paris police still patrolled the streets. Occasionally could be heard the drone of an unmolested 'plane.

The loudspeaker vehicles with instructions to citizens are also decried in various books, from John Williams' The Ides of May: The Defeat of France, May-June 1940 to Herbert R. Lottman's The Fall of Paris: June 1940.

The Gestapo (although in Casblanca there is the possibility that as was common at the time, that Rick refers to the Geheime Feldpolizei, the Gestapo-esque units in the Wehrmacht) did not arrive in the city immediately with the combat troops but took a couple of days until they had established themselves. In the logistical chaos of taking over a city, it is not really possible exactly when they arrive in full strength in Paris but we know that they were there on June 30, 1940 when the first Gestapo agents started to inspect bank vaults in the Paris capital for gold to plunder for the German war economy.

However, the first Gestapo agents as part of the German Sonderkommando, a troop of about 20 Gestapo agents operating independently, arrived in the city together with the troops on June 1940 and established themselves in the Hotel du Louvre. They brought with them arrest lists containing the names of prominent enemies of Germany, communists, socialists, prominent Jews, and government representatives that the Germans wanted to arrest right away and started this campaign in close cooperation with the Wehrmacht commanders on site. But the fact that this Sonderkommando consistent of only 20 people is a pretty good indication that it was the Wehrmacht rather than the Gestapo that drove the loudspeaker vehicles.

They were under the command of Helmut Knochen of the Reich Security Main Office, who later became chief of Gestapo and security for Paris. Knochen was involved in the rounding up of Jews, German emigrés and political opponents of the German rule as well as of the execution of SOE agents and in 1942 was transferred the main responsibility of anti-Jewish policy in France after the Wehrmacht had relinquished it.

The practice of using loudspeaker vehicles to instruct the citizenry of a city that had been freshly taken is one the Germans seem to have employed mainly in open cities, meaning cities where there was the reassurance that no resistance was offered. When there was the suspicion of still lingering resistance somewhere, they rather used poster campaigns after establishing themselves there as far as I can tell.

So, to sum up, the description offered in Casablance is somewhat accurate as in the loudspeaker vehicles existed and were employed in Paris but they were not from the Gestapo but rather from the Wehrmacht itself.

Sources aside those mentioned:

Jacques Delarue: The Gestapo.
Henry Sergg: Paris Gestapo.
Michael Wildt: An uncompromising Generation.

Edit:

On the Geheime Feldpolizei: Klaus Geßner: Geheime Feldpolizei: Die Gestapo der Wehrmacht.