French Restoration is one of my favorite topics, and in many of my timelines it comes about. The restoration of 1870 is very intresting because of the fact the Count of Chambord was very stubborn, but it had more to do with than just the flag. The House of Bourbon was a very conservative, while the Orleanists were more liberal, the Count was wanting to rule as an absolute monarch, he didnt want to have a constitution binding his powers. The battle over the flag just became the ultimate symbol over this conflict, and caused a great divide again between the Orleans and the Bourbons.
The monarchists best hope was the Count were to die, which happened over a decade later but by that time the Republicans were in firm control and the issue never became serious again. There are still those in France that want to restore the monarchy but they have a snowballs chance in hell I think.
If the Count were to accept the throne without battling over the flag or his style of rule then quite a bit could change in European history as Grey Wolf said. Perhaps King Edward VIII marries the daughter of the French King, and has a handful of children. He doesnt abdicate and everything from 1936 onward is changed drastically right there. Maybe another of France's princesses marries into Russia, during the Revolution France feels obligated to help her. Even if the communists still win, French troops manage to save their princess and the Russian Royal Family, before dying Lenin orders a purge of the party, among the victims is Joseph Stalin, Leon Trotsky becomes the new Premeir of Soviet Union.
How this would effect the First World War is beyond me, I dont know if it would, other than giving France a symbol to rally behind, or rally against once things start to go bad.