How can I put my Instagram client into Live mode?

You have get your app reviewed by Instagram first.

Click Edit for your client, Click on the Permissions tab and submit for review. Once approved, u will be able to click Go Live

I put my application from last 12 days in Instagram Developer Console but not getting any reply from them so I found a new way to submit the app using below steps and It's work for me.
Old Process:-


I think there is some update related to the Instagram App review, please check below steps & screenshot you will get the idea of how to submit Instagram app from Sandbox to Live mode in the new updates.

Steps:-

1) https://developers.facebook.com/ Click on this link and enter into the facebook developer console.

2) Now click on My App or Create the new app if there is not any created.

3) Click on Product(+) which is in the left side panel (More clarification see the attached screenshot)

4) Select the Instagram and click on Add button

5) Now you will be able to see one more menu in panel namely "Instagram" (More clarification see the attached screenshot)
6) Click on that and you will be able to see the list of permission in right side view 7) Select your respective permission which you require and submit the app for review.

[Note:- You need to give your .apk and .zip file accordingly android & iOS. You also need to submit screencast of your flow and especially to show the part which indicates why you need this permission]

Since 17th of November, Instagram made his API more protected for spam/fraud/copycat apps. You can read more about it here: https://www.instagram.com/about/legal/terms/api/