Quite a lot of interesting things actually, although this is quite a large question.
Egypt
Egypt was undergoing an era of extraordinary turmoil. In the pre-war decades the major political movements in Egypt like the Wafd and the Muslim Brotherhood held massive street rallies inspired by their equivalents among the fascists and the communists on the continent. Public opinion, given the anti-imperialist sentiments of the population seems to have been pretty clearly pro-Axis. The terms of the 1922 Unilateral Declaration of Independence however (unilateral because Britain declared independence for Egypt, and also dictated the terms of that "independence.") essentially gave Britain power to operate militarily within Egypt with no restrictions and to control Egypt's foreign policy.
The result of this conflict, a generally pro-Axis government/populace contrasted with the actual power that Britain still held in Egypt led to the Abdeen Palace incident in 1942. Basically the British surrounded King Farouk's palace with tanks, demanded he either concede to British demands over the composition of his government or be forced to abdicate.
He conceded, and according to some descriptions I've read he was basically left a broken man. Nonetheless, Egypt remained officially neutral until 1945. However Farouk's apparent subservience to the British was on of several factors involved in the revolution against him in 1952.
Iraq
Like Egypt, Iraq was generally pro-Axis, and to that end Rashid Ali, an Iraqi minister, staged a coup. Like the situation in Egypt, this pro-Axis sentiment was largely tied to anti-British anti-Imperialism.
In response Britain invaded Iraq, deposed Rashid Ali, and occupied the country until 1947.
I have to admit that these events in Iraq and Syria (to follow) are the ones that I have studied the least of the countries I will reply about.
Syria
The Syria campaign, as far as I can tell, was sort of an embarrassment for the French and the allies. After the fall of France, Syria was governed by Vichy forces. The military campaign in Syria was quite brutal considering how little known it is (thousands of casualties,) with French forces led by Henri Dentz, who was sentenced to death after the war (then commuted, although he died in prison.)
Syria would also prove quite awkward for the Free French. Wikipedia, from what I can tell, completely glosses over this but although the Arabs of Syria/Lebanon seemed to understand that the promise of the Free French was independence for Syria/Lebanon, there was, nonetheless, an at reoccupation, including Free French forces conducting a late night raid to arrest the President of Lebanon. The Free French, however, simply did not have the power to carry off such a reoccupation and duly recognized the independence of both countries towards the end of the war.
Palestine
Our resident /r/askhistorians expert on this is /u/tayaravaknin, but some really fascinating things were happening in Palestine immediately before, during and after World War II.
From 1936-1939 there was a major uprising among the Arabs of Palestine against British rule, inspired by the death of cleric Izz ad-Din al-Qassam.
The British response was a classic, if brutal, counter-terrorism campaign. They conducted group reprisals against rebel villages, arrested thousands, carried out executions, and, crucially, started arming and training Jewish militia groups and developing stronger ties with the Jewish Agency.
The atrocities of the Nazis were also driving a new wave of Jews to Palestine. Prior to 1939, hundreds of thousands of Jewish immigrants fled Europe for the Holy Land in the "Fifth Aliyah." This led Britain to attempt to impose stronger immigration restrictions in the White Paper of 1939.
However as the scale of Nazi atrocity grew the need to oppose both Britain and Germany led to conflicts within the Zionist movement. David Ben Gurion, at the outbreak of war, declared 'We will fight the White Paper as if there is no war, and fight the war as if there is no White Paper.'
His use of the word "fight" was not metaphorical. During WWII and afterwards various Zionist groups, most notably the Irgun (Menachem Begin being its most prominent member, and whose memoir "The Revolt" is fantastic.) but with a level of cooperation from the Haganah/Jewish Agency which will probably never cease to be debated, conducted essentially a terrorist campaign against the British to attempt them to leave. This culminated in the bombing of the King David Hotel in 1946.
The Arab factions in Palestine, as in Iraq, as in Egypt, were probably pro-Axis, as infamously exemplified by the actions of Haj Amin al-Husseini, who fled Palestine after the revolts of 1936-39 and eventually ended up in Nazi Germany and Italy. Al-Husseini is particularly nasty figure in the history of this, including apparent complicity in the holocaust, but general Arab sentiment, as elsewhere in the middle east, was certainly more motivated by anti-Colonialism than by genocide.
That's more or less a summary of my knowledge of the WWII experience of those countries, but it's hardly exhaustive. I'd be particularly interested in hearing from someone more knowledgeable about the history of Turkey, as its my understanding that the machinations of the various powers to try and get Turkey onside during the war are fascinating, definitely an area I'd love to know more about.
Source wise, I would check out The Society of Muslim Brothers by Richard Mitchell, Redefining the Egyptian Nation 1930-1945 by Israel Gershoni and James Jankowski, Western Imperialism in the Middle East 1914-1958 by D.K. Fieldhouse, One Palestine, Complete: Jews and Arabs under the British Mandate by Tom Segev, and as always for an excellent and highly readable overview The Arabs: A History by Eugene Rogan.
Palestine
Working off /u/CptBuck's framework, I'll expand on what happened in Palestine as far as the Zionist groups, and the Palestinian-Arabs/Muslims in the area.
I think it's important to note first which Zionist groups were predominant in the fighting during WWII.
Many Zionist groups abstained from fighting, including the most prominent, the Haganah. Led by Ben-Gurion, and around since the 1920s, the Haganah would form the backbone of the IDF when established in 1948.
Ben-Gurion was truly sincere about avoiding fighting with the British during WWII, by most accounts. This was the general policy of the overall Zionist movement and bodies. The Jewish Agency Executive announced on the day Britain declared war on Germany:
At this fateful moment, the Jewish community has a threefold concern: the protection of the Jewish homeland, the welfare of the Jewish people, [and] the victory of the British Empire...The war which has now been forced upon Great Britain by Nazi Germany is our war, and all the assistance that we shall be able and permitted to give to the British Army and to the British people we shall render wholeheartedly.
Irgun, led by Begin and Shamir and those like them, offered a truce. Only the smallest group, and the most extreme, continued fighting: Lehi (Lohamei Herut Israel), otherwise known as the Stern Gang.
Overall, the truce (excluding Lehi of course) held. There were incidents of breaches, however, most often as retaliation for immigration policies the Jews deemed unfavorable. When the British began deporting immigrants (illegal ones) en masse to special island camps in Mauritius and Cyprus, the Haganah tried to subvert the policy by blowing a tramp steamer docked in Haifa. They miscalculated the strength of the charge and ended up killing 252 of the refugees on board.
Still, while the British were suffering defeats and Rommel's armies were at the gates of Egypt, Palestinian Jews rushed to join the cause of the British. 25,000 to 28,000 would join the war on the British side, with thousands more serving as auxiliary policemen, and the British were actually reluctant to actually have them in the first place. Advances into Vichy-controlled Syria and Lebanon were scouted by Jewish forces in 1941 (where Dayan lost an eye), and Irgun assisted the British push on Baghdad (the IZL team's commander, David Raziel, was killed by strafing aircraft).
The Zionists had even pushed, from the start, for a "Jewish army" but a distinctly Jewish brigade was not formed until September 1944, which saw action in only Italy near the end of the war.
The Irgun forces resumed the struggle against the British after they believed it was clear the British would win, on February 1, 1944. Whether it was clear or not is debatable, but at any rate they chose to resume the fighting then officially. Still, the Haganah attempted to dissuade them and stop them. Indeed, from November 1944 to March 1945, the Haganah rounded up and transferred to the British Irgun and Lehi (most were Irgun that got arrested, despite Lehi being more extreme, because Irgun posed more of a threat to Haganah hegemony most likely) forces, usually after beating and interrogating them.
Lehi had fought the entire time, Irgun didn't resume until February 1944, and generally speaking that truce was held. Even Begin's Irgun directives were to attack civilian installations linked to the mandatory authority, not military sites, so it would be less likely to be seen as harming the war effort (unlike Lehi's goals), and the goal was to avoid British personnel deaths that might harm that war effort.
Lehi, on the other hand, offered to ally with Nazi Germany at least twice, but eventually decided to try and ideologically align themselves with Stalin.
Just providing some context on the internal fighting between Jews and the British, but now I'm going to get into the Arab/Palestinian-Muslim fighting in Palestine during WWII:
Palestinians and Arabs were very conflicted. On the one hand, the British controlled the areas they were in, but on the other resentment for the British crackdown in the 1936-1939 campaign (along with the general feeling that the British would lose the war at the outset) led Arabs to feel very conflicted in Palestine in particular.
Khalil Sakakini was not unrepresentative when he noted in his diary, in mid-1941, that the Arabs of Palestine "had rejoiced when the British bastion at Tobruk fell to the Germans. Not only the Palestinians rejoiced...but the whole Arab world, in Egypt and Palestine and Iraq and Syria and Lebanon, and not because they love the Germans, but because they dislike the English...because...of their policy in Palestine."
Still, the Arabs did not outright rebel or fight the British. Part of it was the persistent weakness of Palestinians following the revolt: British policies had, after all, led to a very large amount of young Arab males being killed during the revolt. However, a wave of prosperity and the presence of Allied troops throughout Palestine meant a revolt was virtually impossible, and the Arabs were far less inclined to take any sabotage efforts like Lehi. Hajj Amin al-Husseini even sent agents in 1940 to Palestine to see if a new revolt could be incited, but they saw no opportunity.
Palestinian exiles contributed against the British elsewhere, however. Husseini issued a fatwa (religious ruling) calling on all Muslims to wage jihad on Britain. Some riots broke out in Syria, but generally Palestine remained stable. Husseini continued trying to get Muslims from Albania and Yugoslavia to fight the British, and cooperated with the Nazis in that regard. Husseini even met with Hitler in November 1941 and promised to organize a new pan-Arab revolt, and he tried, but failed.