Intuition or instinct (which is the word I prefer in this case) in the sense of “gut feeling” has a role in the Islamic lifestyle. However, because it is not objectively verifiable, the priority is given to knowledge, and the sources of knowledge are recognized by Muslims as: the mind (i.e. reason and logic), the external and inner senses (emotions), and true information (like Napoleon existed, China exists, and the like). Instinct is rejected if it contradicts any of these, especially if it contradicts any known rules of the religion, but beyond that it is a personal matter. If a person is a very pious Muslim, his instinct may become a source of knowledge for him, so that he senses things that are unseen to others. Again, such instinct cannot conflict with known rules of the religion.
I should mention here that the Muslim concept of “belief” does not have any relativistic connotations. Belief in Islam entails both knowing the truth with certainty, admitting it willingly, and humbly submitting to it in the heart without scorn. Anything less than that is not Islam, and not belief in Islam. That is why epistemology (the concept of what knowledge is and does) is very important is Muslim scholarship.