If you literally mean Franz Ferdinand leaves Sarajevo on June 28th unharmed and Europe stumbles along for a while longer, it could be interesting.
One area that never really seems to get talked about is growing trade unionist unrest in the early 1910s....
In Britain you have the issue of Ireland, the Home Rule Bill had finally been passed in 1914 but the War stopped it being instituted as law...
The 1917 Ausgleich will be messy IMO, and might be a trigger for war.
In Germany I think the Bismarkian Democracy will eventually crumble under public pressure ...
Another major trigger for war, and one that arguably had a strong pull on Britain to enter it, was German encroachment into the Iraqi oilfields. ....
Yep, the "Belle Epoque" was really quite the powder keg, wasn't it. Throw in rising Socialist votes in the USA, bear in mind that "rising trade union unrest" was an issue not only in the developed capitalist nations but also in Russia where it was linked to Bolshevism, Menshevism, and the Social Revolutionaries in an agrarian form, factor in rising anti-colonialism in not just the British Raj, but Vietnam for instance, consider the internal destabilization of Turkey other commentators point out (meaning that the Ottoman regime probably couldn't survive without being shored up by some European power or other, probably Germany implying yet more Anglo-German flashpoints), consider the breakdown of the Manchu Empire and the way the USA kept getting entangled in suppressing revolution not only in Mexico but throughout Central America and the Caribbean...you know, this is why I give so much weight to Lenin and Trotsky's analyses of the basic causes and nature of the Great War. From this perspective it is pretty ASB not to have some kind of Great War or other start sometime between 1910 and 1930 at the latest.
I keep waiting for updates on LordInsane's "Central East" timeline because instead of wishing away the Great War, he seems to have hit on the next best thing for a more "Belle Epoque" kind of world--have the Great War happen but in such a fashion that the necessary social earthquakes that so much pressure has been building up to cause do happen, but do as little damage as possible, leaving the world to move forward with relatively little disruption and with many of these issues either resolved or a clear path for peaceful evolution of them opened up.
To wit: the War breaks out between France and Germany, but France invades Belgium first, thus alienating the rest of Europe. Britain and the USA never get drawn into the war at all--meanwhile Britain resolves the Irish issue peacefully. Germany concentrates on defeating Russia first (until the French invasion of Belgium anyway); Russia falls a year early, the revolutions there happen more along traditional European bourgeois lines. Actually they are strongly leftist, but a coalition of leftists that avoid the one-party state basis of Soviet Red terror OTL, meanwhile the Red/White civil war happens sooner, quicker, and leaves the Red state based in Moscow shorn of much territory, but with more of its people left alive and more industrial base left intact, so the leftist state develops more along the lines that socialist idealists hoped it would and also cultivates better relations with the capitalist world. Germany seizes most French colonies (but not Algeria); the Ottomans are allies of victorious Germany and so are upheld. In Germany, the leftists continue to operate in the quasi-parliamentary Second Reich framework and so presumably there is a chance of peaceful evolution of a more liberal society there, I gather this also has a good influence on inclusive progressivism in the Ottoman sphere.
There are still lots of long fuses burning here of course. How will any of the capitalist powers handle domestic trade unionism/socialism? Will they accommodate worker's parties in the parliamentary system, allowing peaceful evolution of something like the New Deal social balance of power, or is some sort of bloody confrontation inevitable (perhaps, as Lenin/Trotsky claimed for OTL, diverted instead into imperialist war, meaning another round of Great War?) What of the rump Red Russian state confronting various White regimes that surround it--will the latter become tinpot oligarchies practically begging for Red intervention? Or will any of them become strong right-wing, more or less fascist regimes that pose a serious threat? How strongly will the bourgeois parties in Germany, Britain, the USA, Italy wish for someone to strangle these pestilent Reds that set such a subversive example for Western workers, insofar as they are successful domestically--and if they aren't successful domestically, will the Red Russian regime inevitably fall into something like Stalinism anyway despite their better start? Would such a Stalinist regime be strong enough to fight off those who want to feast on its corpse? What of the colonial issues--can people like Ho Chi Minh be recruited to support a more enlightened colonial regime that evolves toward inclusive democracy, or will the Germans running Indochina make the French OTL look like saintly philosopher-kings by comparison? How is Britain going to react to a German High Seas Fleet that has legitimate business in peacetime steaming past them to bases in formerly French Africa and Indochina, a fleet that will presumably grow and modernize further? What bearing would this allegedly peaceful rivalry between Britain and Germany have on their colonial policy--would they go for better relations to secure their hold on India by co-opting more Indians into the Raj, or panic and clamp down harder? What of race relations in the USA and what bearing would that have on the development of US relations to Latin America and for that matter, by example and analogy, British and German colonial relations in Africa?
Sorry to wander so far from the topic of "No Great War at all." I think this is so improbable, for reasons outlined above, that it is up to whoever proposes it to work out how it happens that the huge military forces mustered by the great powers never do get actually used on full scale. Just as a general proposition it seems wildly unlikely to me.