From Stephanie Hertzenberg at Beliefnet: Are Islam and Evolution Compatible? The question is controversial and hotly debated.
Helpful information but some well-meaning misdirection:
Some Muslims hold that evolution is partially compatible with Islam. At “Have Muslims Misunderstood Evolution,” a London event organized by the Deen Institute in 2013, Shaykh Yasir Qadhi argued that Islam is compatible with all of Darwin’s theory of evolution except in the case of humans. He claimed that, from an Islamic theological perspective, a Muslim can say that Allah inserted a created Adam into the natural order. This would be, he explained, as if Adam were the last domino placed in a line by Allah. Non-believers would see Adam’s domino as a casual connection or continuation from all the previous dominos, but faithful Muslims would see Adam’s existence for the miracle it was. This position allows for a person to accept most of modern science but also preserves the miracle of Adam’s creation from a theological perspective.
At the same event, Ehab Abouheif took a slightly different stance, but did argue for at least the partial compatibility of Islam and Darwinian evolution. He claimed that the facts of evolution cannot be disputed. There is, after all, ample evidence of evolutionary processes. Abouheif did say, however, that which processes took place during an organism’s evolution and how those processes took place are up for debate.
So far, so BioLogos. We are not yet at the point of arguing that human beings are meat puppets. But then we read,
In Islam, just as in many religions, the debate over the compatibility of faith and science has raged for decades. Some Muslims reject evolution entirely. Others accept it wholeheartedly. Given the size of the worldwide Islamic community and the diversity of Muslims everywhere, there is unlikely to be a consensus on the issue anytime soon. Instead, every Muslim will have to decide for themselves: can they believe in both Islam and Darwin’s most famous theory? More.
Ah, a question we can answer. In short, no.
Longer version: You can’t believe in Islam and in Darwin’s theory because Darwin’s theory, as he himself knew*, rules out the idea that there is any kind of truth that the human mind is capable of apprehending. Major Darwinians like Daniel Dennett, fearing little resistance from the postmodern world, make that clear.
So if you believe that Islam is true, you can believe evidence-based claims about the history of life (dinosaurs and all that). But not Darwin’s theory, which is explicitly recognized by adherents as naturalist atheism.

*Tales from the quote mine:
“But then with me the horrid doubt always arises whether the convictions of man’s mind, which has been developed from the mind of the lower animals, are of any value or at all trustworthy. Would any one trust in the convictions of a monkey’s mind, if there are any convictions in such a mind? – Charles Darwin
[To William Graham 3 July 1881]”
Darwin could afford to ignore the problem, possibly because he knew that his followers would twig to the fact that just shutting down discussion, legislating curricula, ending careers, etc., can stand for a long time in place of a reckoning. Most people are busy. Most science teachers just want to teach what the book says. Most science writers are not curious people; they are too busy congratulating themselves on Correct Thought. If you want to think for yourself on the subject, keep abreast of the news but stay out of the swamp.
See also: Why I am not a young Earth creationist
New atheists and the left fall out over Islam
and
The illusion of consciousness sees through itself.