Even more, exactly what kind of an modern Ottoman Empire are you thinking of?
Is it the classic Ottoman Empire as it was around 1800-1820?
Or are thinking more of a reduced empire, in which the main non-muslim area's like Bulgaria, Macedonia, Serbia,
Armenia, Greece (or perhaps several Greek states, like Pontus and perhaps Cyprus) have become independant?
Remember that in OTL, the Empire imploded mainly because of ethnic and especially religious tensions.
How would a modern Ottoman Empire cope with that?
And then there is the issue of the role of Islam and the Caliphate (which was claimed by the sultan)...
How will these affect the world, and especially the muslim world?
Remember that until the fall of the Ottoman Empire, several politicians and researchers from the Western colonial powers feared the possebility that the Ottoman sultan would use his official position as Caliph to gain influence over the millions of muslims in the European colonies, like Africa and south Asia.
And in the WWI in OTL, the sultan/Caliph actually used his authority to declare the jihad againest the Allies,
hoping to start an immense rebellion in the vulnerable French and British colonies, as well as in Russian Central-Asia.
The attempt failed miserably, though, as almost no muslim actually started an uprising or any other form of hostile action to their colonial government. Most muslims also saw that this jihad was purely political, and had nothing to do with actual fighting in the name of Islam.
And it is worth mentioning that the Ottoman minister who actually proposed this plan, was in fact an atheist...
And because of that, all those good old encyclopedias from the fifties and the sixties described the jihad as a dead letter, perhaps comparable to strict fasting and the modern Roman Catholic Church.
Nonetheless, the survival and continuation of the Caliphate would still have a notable effect on sunni islam in general, especially in world politics in times of peace.
And it would also make quite a difference if the cities of Mecca and Medina remained into the hands of the Ottomans
and the moderate Caliphate, instead of a fundamentalist wahhabi-kingdom, as we see today with Saudi-Arabia.
And as an interesting footnote: in case the Caliphate would be maintained,
what kind of effect would it have on modern often wahhabi-inspired muslimfundamentalism and political Islam?