Assalāmu ʿalaykum, Ustadh! Is it permissible to recite " Lā ilāha illā Allāh " with the mouth closed and without moving the lips, especially while riding a bike or in public? Since it contains no letters articulated by the lips, must the mouth still be opened for the dhikr to be valid?

Ustadh AbdulAziz al-Haqqan

This is invalid. Yes, you must actually say it, even if it is without speaking out loud; you are still saying it, and air is coming out of your mouth.