Well the war began to move into the middle east in 1941. Firstly in Iraq.
Iraq at this time was at this time a monarchy of Faisal II under the regency of his uncle prince 'Abd al-Ilah. The country was at this time heavily under the influence of the United Kingdom which maintained a number of military bases there and significant stakes in the country's oil economy. The Germans therefore encouraged a group of Sunni Arab Nationalist Army officers to begin a coup in Iraq early in 1941 in order to drive british influence out of the country in support of the Nazi's efforts at this time in the western desert campaign. The Nazis heavily encouraged the move but didn't actually send any significant force to fight there, only engaging in some air skirmishes with the RAF.
Although the coup was initially successful in driving the government out of Baghdad Britain quickly responded and with the support of monarchists and ethnic minorities such as Assyrians restored the country's monarchy and occupied the country.
Meanwhile just to the north prior to the war The Republic of Syria and Greater Lebanon had since the first world war been within the French Sphere of influence and was effectively administered in part by the French and French educated allies as part of their Mandate for Syrian and the Lebanon. When France fell in 1940 the French forces in the region aligned themselves with the pro-Fascist Vichy government. Later in 1941 British and Free French forces launched an offensive into the region. Again the Nazis were involved almost entirely in the air but in this but this time on a larger scale. But the allies won the engagement and Free French forces took over the influence the Vichy ones had in the region.
Then there is Iran. Iran was nominally neutral at the start of the war. But as the war in the east started in 1941 the need for a corridor for goods to be able to supply the Soviet union came up increasingly necessary and Shah, Reza, of Iran who had some Nazi sympathise refused to allow goods to move through Iran. So Britain and the Soviet union launched a brief invasion of the country which forced Reza to abdicated in favour of his son and forced Iran to allow the transit of good from the Gulf of Persia to the Soviet Union by a rail and road corridor.
Nazi agents would attempt to disrupt the corridor by attempting to insight unrest among the people of the country which forced the allies to make concessions concerning their occupation but didn't in the end result in the destruction of the corridor.