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Open the Facebook app on your iPhone or iPad. Tap the ☰ icon. Scroll down and tap Settings. Tap Account Settings on the pop-up menu. Tap Security and Login. Find the WHERE YOU'RE LOGGED IN section in Security and Login. Tap the ⋮ icon next to your Messenger session. Tap Log Out.
The Facebook app looks like a white "f" in a blue square icon on your home screen. The Messenger app doesn't allow you to sign out. You'll have to use the Facebook app to sign out of your account on Messenger. This button is in the lower-right corner of your screen. It will open your navigation menu. This option is at the bottom of the menu. A menu will pop up from the bottom. This will open your account options on a new page. This option is at the top of the Account Settings menu. This section lists all your active sessions, including all mobile and desktop Facebook and Messenger logins. Find the Messenger session you want to log out under WHERE YOU'RE LOGGED IN, and tap this icon next to it to see your options. This will log you out of your account on the Messenger app.