As in the title, I currently have Elective Monarchy as a Muslim Sultan and am wondering whether my succession law will automatically change to Open if I switch to Limited Council Authority.
No, the Law will remain Elective unless you change it manually to open. From the CK2 Wiki:
If Muslim or Eastern, (or Bön???) must have Conclave DLC and the council law "full council authority" to enact elective succession. However, elective succession does not become invalid after you revoke council authority, so the law will not revert after succession.