I'm experiencing similar issues. I only see this when using the Dish Anywhere App on my new FireTV. I do not see this on two other TV's where I'm using Dish Anywhere App via a Fire Stick.
I do have a work around.... If I turn off my TV's built-in speakers and use Bluetooth speakers instead, then the lip sync issue goes away. I also tried using a soundbar connected via an optical cable, but the lip sync issues remained.
We definitely need to get this issue over to our engineering department for further review. Can you send me a private message with the phone number and 4 digit security code on the account so we can begin working towards finding a resolution?