The text notifications that Instagram sends to your mobile device are actually push notifications and not SMS messages. Push notifications are set by default to send a message to you when someone likes or comments on one of your photos, mentions you in a comment or when one of your photos is listed on the Popular page. You can discontinue push notifications in your account settings to stop Instagram from sending these messages to your device.

Disable Push Notifications

Open your profile page in the Instagram app on your iPhone or Android device. Tap the “Settings” icon and select the “Push Notifications Settings” option. Tap each notification type to disable. On Android devices, you can disable Like Notifications, Comment Notifications, Contact Notifications, Photos of You and New Follower Notifications. On iOS devices, you can disable Like Notifications, Comment Notifications, Contact Notifications, Photos of You and Direct Notifications. You can also disable all notifications for the Instagram app in the iOS Notifications Center.