There are many who say it has, particularly with the Israel-Palestine conflict. Many say that the conflicting promises the British made, for example, to the French, Russians, Jews, and Arab-Palestinians are the cause of the conflict around Syria, Lebanon, and Israel (and even Egypt, to a degree). It's certainly possible to argue that Iraq to some degree was destabilized because of the British control there for a time, though that's not my specialty.

What you have to keep in mind, though, is that this is only one facet of a larger problem. The destabilization of the Ottoman Empire certainly facilitated growth of national identities (and conflict, in some areas), but many scholars also argue (again, drawing from the Israel-Palestine conflict, though authors in general tend to agree from what I've seen on other country topics) that the national identities were already growing quickly amidst the decentralized and ineffective Ottoman empire administration. Plus to take the tack that this process is all the fault of the Europeans, instead of understanding the conflict as a wider amalgam of European influence, conflicting ideologies, wartime decision-making, foreign intervention, and Islamic Sunni-Shiite infighting, is to ignore the nuanced factors that contribute to conflicts around the world.

Consider the conflict of Israel-Palestine. Certainly one could say the conflict reflects today the divisions it always has, and which came about before the Ottoman Empire collapsed: that of Zionism-Palestinian nationalism and pan-Arabism (both are factors). It's unfair to attribute the problems that persist to only this, or even just the British. The problem also lays in part with the Ottomans, who ineffectively addressed the question of Zionism as it arose. Or you could disagree, saying they did attempt to address it, and lay blame with the Zionists who manipulated the British. Or you could say that it was none of this, that the conflict was the fault of the Nazis who invigorated the Zionist cause by creating more immigration to Palestine, which gave the Zionists more of a boost in stature and led to the White Paper, which created more conflict over immigration. Or you could say it was the fault of Faysal Husseini, for attempting to create a pan-Arab state without getting explicit-enough assurances from the British, while also creating an agreement with Zionist leaders that both sides reneged on.

And what if I treat each war as separate? Sure, the 1948 war was begun because of (insert reason here), but 1967 could be regarded a totally separate issue, removed from the question of Palestinian natives and focused more on pan-Arabism. Or is it anti-Semitism. Or is it fear of aggression by Israel. Or is it a struggle by the West and the USSR for dominance in the Middle East. Or maybe it's just the French, who supplied most of the weapons, vs. the USSR. Or maybe it's the Jordanians, who wanted to create a pan-Palestinian state (a contentious term, I know).

See, the problem with sentiments like "The Sykes-Picot Agreement and other European border-drawing destroyed the Middle East and set it up for a century of war" ignores many other factors at play, not least the interplay of religious, cultural, economic, and psychological discussions. Iraq is no more or less complex in that sense; neither is Syria, Lebanon, Egypt, Tunisia, or any other Middle Eastern or otherwise in-upheaval state right now. Surely one could say it contributed, but to say it was the set-up is to ignore that there had to be other factors there with it to set it up. If there were no other sources of conflict, then it's conceivable to say that the Middle East would've simply talked it out, created a pan-Arab state when the British, French, and other powers left, and been just fine with it (with maybe the exception of Israel, but they wouldn't have had such varied aims when fighting Israel if not for their conflicts, so there's that). That's not to say the Middle East is violent, but there are other things at play that cannot be predicted, which cause conflicts around the world. We cannot think of this as unique, when we have so many examples of underlying factors leading to conflict over the past century: World War I, II, Korea, Vietnam, Cambodia, Bosnia, Rwanda, Colombia, China, the USSR...to name but a few. Just as I would frown on someone saying the Treaty of Versailles caused WWII, I would frown on the statement above, and so would almost every scholar I've read if they didn't think it was saying "It contributed to the set-up".

But then, that's just like, my opinion man :).

I think /u/tayaravknin has given a good overview for Sykes Picot vis a vis the Israeli conflict but unfortunately I don't actually think that conflict is a great example of what people mean when they say they talk about the damage from Sykes-Picot.

Part of the issue is that "Sykes-Picot" is often used as a short hand in the popular imagination for European re-drawing of middle eastern maps maps but actually Sykes-Picot was just one of several agreements that hammered out what the post-war world would look like and was not, in any way, an agreement on final status. Moreover the mandate system didn't necessarily determine what the political systems of the mandatory countries (namely, for Britain, Palestine, Trans-Jordan and Iraq, and for France "Greater Syria" including Lebanon) would look like.

Iraq is a perfect example of both issues. Under Sykes Picot Iraq doesn't actually resemble the modern borders at all. That's because an area roughly contiguous with the Ottoman Vilayet of Mosul was allotted to French control under Sykes Picot, but what to do with with that territory became known as the Mosul Question.

The Mosul question is a fantastic example of where imperial interests, map drawing, national politics, demographics and conflict all intersect and it has been ever since.

Sykes Picot gave the territory to the French in part because Britain wanted a French buffer between itself and Russia. If there was any doubt that Sykes Picot is not the final plan the allocation to Russia of Armenia and Istanbul should put that idea to rest. Over the course of the war, however, Mosul was occupied by British troops and administered from Baghdad. Moreover it was widely believed that there was oil there.

But Mosul was also an important part of the political calculus of forming "Iraq." An Iraq that consisted of the Vilayets of Basra and Baghdad would have been about 80% Shia. After the failure (and expense) of direct British rule on the lines of India Britain adopted the "Sharifian Solution." The Hussein McMahon correspondences had promised a large portion of the Arab world to the Hashemites. Prince Faisal had been removed from Syria and it was decided that he, a Sunni Arab with a Sunni Arab officer corps and staff, should be given Iraq as a kingdom.

Faisal wanted Mosul to be a part of Iraq because the regions Sunni Kurds would push the overall Sunni population of the country to about 40% making Sunni rule acceptable if not exactly comfortable. The British, on the other hand, knew that Kurdish rebel movements would force Faisal to keep them in the country as he would be reliant upon British airpower. They also, of course, no longer had any need for a buffer against Russia as the Bolsheviks were not granted any of the territories that had been promised to the Czar.

How much the Iraqis might have chosen this for themselves is an open question. The British rigged a nationwide referendum to choose Faisal as king. But there were some native movements for the creation of an Iraq that would have looked similar. There was pre-war revolt by a Basran notable to form a state out of Basra and Baghdad Vilayets. Faisal's officers were mostly Iraqis and they viewed Iraq maximally as including Mosul as well.

The Shia had a much less coherent vision for any sort of political entity but they did launch a major revolt against British rule in 1920. It also obviously seems unlikely that they would have, of their own accord, chosen a Hejazi prince to be their king.

That said, Faisal himself was popular, although not nearly as popular as he was in Syria where the French had driven him out.

I don't want this description to go on too long but to summarize the answer to your question in regard to Iraq: the role of European imperialism in the modern problems of Iraq is highly emphasized by historians. But to put it down to "map drawing" and particularly Sykes Picot ignores how and why the actual borders were drawn. The people making these decisions, whether in London or Baghdad, had sophisticated reasons for doing so. They cared about things like ethnic and sectarian balance, about the cohesiveness of territory, about the legitimation of native rule, etc. It just didn't work.