The question is rather why Louis Napoleon did step back from his promise to support Sardinia against Austria.
The most likely reason is that (for once) Austria does not behave stupidly, like they did IOTL when they issued an ultimatum to Sardinia to force the to demobilize the army. If there is no ultimatum, it is still possible to avoid the war and let European diplomacy act.
This would not contribute to solve the Italian question in any way, since the political situation in the peninsula is already well above the boiling point. I'd expect that there might be significant insurrections in Italy in the short term (the Papal Legations and the duchies of Parma and Modena come to mind), and Austria would respond as usual: censorship, trials and military actions to prop up Modena and Parma.
From the point of view of Bismarck, an alliance with Sardinia would still be a good solution: he needs to get the Austrians busy on their southern border and Sardinia together with the unavoidable insurrections would be good enough for him.
The French emperor would not be upset anyway: in 1866 the German unification is still a pipe dream, even less feasible that it was IOTL since Italy has not unified. Austria will have blood on her hands, like Russia IOTL, and this would make her the target of dislike of the liberal press (in UK and France mostly). So Louis Napoleon would not make any move beyond what he did IOTL: try to bargain with Bismarck for compensation in exchange for his neutrality. Russia would still be in the Prussian corner, for the same reasons they were supporting her IOTL.
If anything, the non-war of 1859 may have the result of butterflying away the massive failure of the Prussian army during the mobilization of 1859. This might mean that there will be some kind of reform of the army, but it will not address the mobilization issues and the massive use of railways (which was tested in the 2nd Danish war IOTL) and may not even deal with the new infantry tactics (which ultimately were originated by the massive losses of Solferino). To compensate this, there will be no reform of the Austrian army either (it wasn't a great success IOTL, but at least the artillery pieces were mostly updated, and the artillery tactics improved).
The last butterfly might affect the health of Cavour: the years between 1857 and 1861 were very stressful for him, and ITTL some of the troubles might be butterflied away once Louis Napoleon does not intervene. It would probably mean that he would resign and possibly leave active politics for a while. OTOH his health might improve, and he could be back in 1861 and still have a few years on the scene.