Well, in general you might end-up with a situation where France and Austria/Austria-Hungary develop better/closer relations in concern over Prussia.
So, here's a little scenarios;
It's the 1850s and France, under a the House of Orléans, has a youngish, liberal King who knows how far he can go and what's overstepping his boundaries and causing the potetnial for Revolution.
The King has developed some ideas similar to what Napoleon III did IOTL, that is that France should be a Constitutional Parliamentary Monarchy and the King's position should be that of a more or less politically neutral ruler who primarily reigned but also actively tries to ensure the continued growth and progress of the country, essentially hands off apart from exceptional times or in the name of the country.
Under this King France stabilizes politically somewhat and enters a period of economic and industrial growth.
The Crimean War comes and goes more or less as per OTL, though with France developing more of an interest in the OE afterwards.
In the late 1850's the threat of Prussia starts to become apparent, leading to Austria and France developing closer relations.
Sometime in the first half of the 1860's Italian Unification becomes a major issue, however it poses a dilemna for France; on the one hand they don't want to endanger their developing relations with Austria on the other hand they want to improve their relations with Sardinia, ultimately a compromise is found, though not everyone is happy with it; France supports Sardinia in their conquest of northern and central Italy, excluding Lombardy-Venetia and the city of Rome and its environs, and in return is ceded most of the County of Nice (slightly smaller than OTL).
Essentially this created a situation in which Italia is split between the Kingdom of the Two-Sicilies in the South, the Kingdom of Sardinia in the Northwest and Central Italy, the Independent State of Rome and Austria in the Northeast.
This cools tensions and calls for unification for the time being and allows France to focus on other matters.
As a result of the butterflies from their relations with France Austria and the South German states don't support Prussia, leading the war to be seen as one between Europes two 'Armies with States', rather than a German war.
Tensions remain high in the 1860's, with Prussia seeing itself as the true power of the German lands, eventually leading to a dispute and ultimately war sparked by a Pan-German uprising in Baden in the 1880's with Prussia and the North German states on one side and France and Austria on the other and Sardinia wondering if siding with Prussia might just allow them to fulfill their goal of Italian Unification.