Instagram does not allow you to turn off read receipts, but you can read messages privately if you first turn on airplane mode. However, you need to avoid tapping on new message notifications first.
TURN OFF INSTAGRAM MESSAGE NOTIFICATIONS
The message you tap gets marked as read, and you can’t undo it. You can prevent this by turning off Instagram Direct notifications. Here’s what you need to do.
1. Log in to Instagram and open your profile.
2. Tap on the menu icon (three vertical lines).
3. Click on Settings.
5. Select the messages you want to send. (There may be a Direct Messages section.)
6. Click the Off box next to the Message Requests and Messages tab.
READ INSTAGRAM MESSAGES WHILE IN AIRPLANE MODE
Before opening your inbox and tapping any messages, turn on airplane mode.
1. Open the Instagram application.
Tap the Messenger icon in the top right corner. Click the mail icon to open the Instagram Direct inbox if you haven’t switched to the merged Instagram/Facebook Messenger inbox.
3. Make sure Airplane Mode is on and Wi-Fi is off.
For Android, swipe down from the top of the screen to access the Quick Settings menu. Tap Airplane mode.
On an iPhone, swipe down to open Control Center and tap Airplane mode.
4. Take a moment to read your messages.
5. Exit your inbox and return to your profile.
6. Tap on the menu icon (three vertical lines).
7. Click Settings at the bottom of the screen.
7. Scroll down and click Log Out.
9. Switch off the airplane mode.
10. Log in to your account again.