Austria minus Bohemia would have a very hard time waging war on anyone. Cisleithania (and therefore the Habsburgs) would be much weaker than Transleithania and the powerful Hungarian lobby that was against intervening in the Franco-Prussian war. More than that, the loss of Bohemia, its coal, and all its human resources would effectively neuter Austrian industry. The economic fallout of this would be brutal, to say nothing of the empire's much-reduced ability to make the weapons and equipment necessary for all the post-Königgrätz reforms.
This would also be big for Germany too. Would Bohemia have been annexed into the Kingdom of Prussia proper, or "liberated" into the German Federation? Judging by their treatment of the Poles, I doubt the Prussians would be overly tolerant of Czech nationalism.
While I think you could reasonably escalate the Franco-Prussian War into a much larger conflict, I'm not sure you'd get a truly WW1-style conflict (if that's what you're going for). The financial and military innovations needed to make a war that huge last that long hadn't become standard across Europe yet, so you'd probably get a shorter conflict with outcomes that aren't quite as dramatic as the Paris Peace Conference in 1919.