After some troubleshooting, It looks like the issue of renewing subscriptions only occurs when you are using the payment gateway.
Issue: When a member goes to renew a subscription, they type in the CC information and everything process' with out any issues. However, the membership stil says expired (even though the date changes to reflect the new expire date in the future.
I have done this with all plugins deactivated, and using the 2015 theme to ensure no conflict issues. If I use a coupon to renew an expired membership, the issue is still present as well. So it seems that the payment gateway may not be the issue since no payment is even collected when using a coupon.
I have created a troubleshooting environment here - http://projectr.staging.wpengine.com/ - As you will see, there are no plugins activated and the theme is set to default. Please feel free to do what ever you want in the staging environment. Here is the doc for the stripe testing page - https://stripe.com/docs/testing - This will have Test credit cards etc that you can use. Feel free to create new accounts and all that. I have enabled the support option.
I am more than happy to provide login credentials as well for a user that has an expired account (like the one with the screenshot below). Thank you so much and let me know what I can do to help the issue as well.