Use of the ASP.NET view state can cause issues within a multi-server environment.
Resolving view state message authentication code (MAC) errors
View state is information that is round-tripped between WebForms (.aspx) pages in an ASP.NET application. ...
Because the __VIEWSTATE field contains important information that is used to reconstruct the page on postback, make sure that an attacker cannot to tamper with this field. ...
To prevent this kind of tampering attack, the __VIEWSTATE field is protected by a message authentication code (MAC). ASP.NET validates the MAC that is submitted together with the __VIEWSTATE payload when a postback occurs.
There are a number of approaches to mitigate this issue with a multi-server environment (elaborated within the article linked above).
Do these approaches need to be applied within an Azure Web App that is set to auto-scale?
According to your description, I checked the document you mentioned about resolving view state message authentication code (MAC) errors.
As the case about the web application is running in a farm (multi-server environment) states as follows:
Some web hosting services, such as Microsoft Azure websites, take steps to synchronize each application's auto-generated key across their back-end servers. This lets applications that have not specified an explicit element to continue working in these environments, even if the application is running in a farm. If your application is running on a third-party hosting service, please contact your hosting provider to determine whether this situation applies to you.
Also, I tried to create my ASP.NET WebForm application and deploy to Azure Web App (multiple instances), it could work as expected.