Turning off auto-play in your browser will not have any effect in your mobile app usage. If your biggest concern is saving on your data costs, make sure to follow the instructions for your Android or iOS device.
If you want to turn off video auto-play on Facebook in your browser, go to Settings - Video - Auto-Play Videos and select Off from the drop-down menu.
For iOS users, launch Facebook on your iPhone and tap More - Settings - Account Settings - Videos and Photos - Auto-play and select Never Auto-play Videos. If your only worry with auto-playing videos is data usage, you can also choose On Wi-Fi Connections Only.
Android users tap the hamburger/menu icon > Autoplay > Never Autoplay Videos.
If you want to turn off video auto-play on Twitter in your browser, go to Settings - Account and scroll down to Content. Under Video Tweets, make sure Video Auto-play is unchecked.
On your phone, go to Settings - Data - Video Autoplay and check Never play videos automatically.
With Instagram, you can only do this using the mobile app. Go to Settings (go to your profile page and on iOS tap the cog icon, or on Android tap the icon with three dots) - Cellular Data Use - Use Less Data.