I assume you are referring to the War of the Fourth Coalition.

Prussia went to war with France even after being handed over Hanover (no pun intended) because Prussia took a general dislike of French Dominance over Europe. France had been the thorn in everyone's side for centuries (ie Thirty Years' War, WOSS, and others), and they just got sick of them. Prussia fought and lost, pretty badly. They had a recent reputation of glory, and they used it all up after FTG died.

Russia went to war with France because they also took a dismissive view over French Dominance. The thing is, Russia didn't have much to lose, unlike Prussia. Russia is so hulkingly massive and far away, they most likely thought that Napoleon would not be able to inflict a wound too deep if it ever came on their land. Turns out, it did, but only for a small amount.

When Prussia, Russia, and France signed an armistice, Prussia was basically the main loser. Tzar Alexander and Napoleon actually got along quite well after the war, and Russia even gained some previously Prussian Land.

I hope this explains your question to some degree.

I hope that I am not sounding as too critical of Napoleon. I view him as an influential world leader but not a "giant amongst pigmies" as many of his worshippers in regarding his capabilities.

Adding to the previous comment, whenever Napoleon imposed a peace treaty after his victory, the terms are extremely destructive to the losers. A theory had been put forth that many of Austrian declaration of wars against France is primarily motivated by the loss in the Treaty of Campo Fomio, negotiated by Napoleon. Further losses in the War of the Third Coalition and others motivated them to keep on declaring wars on France until Napoleon is exiled to St Helena.

To look at what behind Napoleon decisions regarding all the treaties, the key components in Napoleon strategic thinking is in fear and self-interest. Napoleon love launching a decisive blow, exterminating the enemies' will to fight via annihilation and terror. Indeed, there are many of the accounts of devastating populace and terrified nobles. However, the terror also create hatred. The moment the wars ended, the rest of the countries in Europe started reforming their armies.

After the Fourth Coalition, Prussia lost half of its territories and its status as a great power. The atrocities in Germany commited by the French armies at the time became a motivating factor that many of its military officers rather joined the Russian rather than working alongside the powerful Grande Armee. I. e. Clausiwitz.

Clausiwitz' s book "On War" described his theories of wars in general. He said that "Wars rarely end". They composed of inherent humans emotions such as "fear and hatred", the creativity (chance and probabilies) of the general such as "Napoleon" and politics (reasons). Napoleon is great at all three factors, yet he reasoned wrongly that he can end his enemy via victories on the battlefield alone.

From 1805 to 1815, Napoleon had been fighting wars throughout Europe. Everytime an army is in your areas, your people are prone to die due to starvation. The Grande armee were conscripted from all areas, the French emperor had authority over. The people of Germany, Austria and Prussia had been more than willing to fight another war. For England, Russia and Sweden, there are the issues regarding trades.

He knew that British navy is too strong, so he waged an economic blockade in Europe thinking it can bankrupt England. Every enemy he defeated on the mainland, and every ally must adhere to the blockade. This is a massive mistake on grand strategy.

The theory regarding trade wars and political economy was in its infancy and probably shaped by this decision more than anything else. The entire economy in the countries of Europe suffered, including the coastal cities of France. The blockade created black markets and smugglers. Many administrators, even ones appointed by Napoleon such as Bernadotte and Murat or even his own brothers, prefer to take a blind eye toward them rather than enforcing it. They might even took steps to protect the merchants from Napoleon' agents.

If Tsar Alexander continued with this policies, he stood a chance of getting assassinated by his own nobles whose wealth were dependant on the trades. Rather than risk facing Napoleon Grande Armee alone, he seeked allies with the British and Swedish. Napoleon invaded Russia in 1812 to enforce the failing strategy and maintain his authority.

In 1813, after the Grand Armee is defeated in the retreat from Russia, the War of German Liberation began. The purpose of Austria, Prussia and Russia is to recover what was lost in previous treaties and reassert their prior dominance, along with taking revenge on French troops for the action in the previous wars. Sweden' s crown prince bring up the importance of trade as the cause for the war in the declaration of war in the letter Napoleon (Swedish Pommeria were invade by French troops on the prince' s birthday because of its trade with the English), while his other purposes is to recover Swedish Pommeria and Norway in order to compensate his Swedish government. England for ending the economic blockade.

Sources:

Owen Connelly. " Blundering to Glory".

Suzanne Desan. "Living the French Revolution and the Age of Napoleon".

Clausiwitz. "On War".

Fredrich Wilhem-Wildburg. "Bernadotte".