The vast majority of the Muslim world was colonized by the Allied powers, so the opinions of the people living there didn't matter too much, since the colonizers generally kept the lid on during the war (not always though; we'll come back to this). There is evidence that the anti-Semitism of the Nazis appealed to certain political leaders in the colonized Arab world, but any definitive statement on this is impossible due to the confounding factor of the Nazis being anti-British and French as well, who were the colonizers and thus understandably disliked by the colonized.

There are three exceptions: Syria/Lebanon, Iraq and Turkey.

Turkey is a very interesting country in the context of World War 2, since they had fairly recently fought and won their War of Independence against the Allies of World War I, including Greece, and had been allies of the Germans during the first war. So naturally, they were not well-disposed towards the Allies. But the Turks also greatly feared a Soviet invasion. As the war between Germany and the Soviet Union escalated, the Turkish military began to regard war with the Soviets as inevitable. As a result, Turkey was in a pretty precarious position, and they walked the line quite adroitly. Overall, in terms of the personal sympathies of Turkey's ruling class, they lay more with the Germans than with the Soviets, but that's not saying much. And the Germans didn't realize much, if any, benefit from this, except perhaps the knowledge that the Turkish border with Bulgaria and occupied Greece was secure.

Iraq underwent a full-blown coup in 1941, in which the British puppet monarchy was overthrown by prime minister Rashid Ali al-Gaylani and four senior officers of Iraqi army, which declared war on the British. The war lasted less than a month, with an Allied victory. A few German and Italian aircraft participated on the Iraqi side. As to the opinions of the people, I cannot say.

Syria/Lebanon is also very interesting. In 1936, the French granted autonomy to the Syrian Republic, which had the right to maintain its own armed forces, but was required to permit the French basing rights for their army, navy and air force. In 1940, the region fell under Vichy suzerainty after the Second Armistice at Compiegne, and the famously mercurial Vichy Admiral Francois Darlan signed the Paris Protocols, which gave the Germans substantial basing rights in Vichy colonial territories in Asia and North Africa, including Syria. Indeed, it was through these agreements that the Germans and Italians were able to basically smuggle aircraft into Iraq for the aforementioned brief war. During this time, British commanders in Egypt and Palestine took no action, for a number of reasons, including distrust of the Fighting French, and concerns about overextension at a time when the Italians were invading Egypt, and when German military power was waxing in the eastern Med, due to the fall of Greece. The British and Fighting French finally attacked on 8 June 1941, when British reconnaissance planes spotted German aircraft at Syrian airbases, and attacks on them were authorized. During the five-week campaign that followed, which ended in an Allied victory, the armed forces of the Syrian Republic and Greater Lebanon fought with the Axis forces, and specifically the Vichy French authorities, who controlled the war with limited assistance from the Germans and Italians. However, this does not mean that the people of either Syria or Lebanon supported the Axis, as with Iraq, and likely reflected the rhythms of colonialism more than any great feeling either way.