I have been noticing this for a few months, Norton 360 when VPN is turned on drains your iPhone battery all the time. More than 50% of the battery consumption all the time is by consumed by Norton. Also I don't have SMS protection enabled. I will try disabling VPN from now on as I don't want to charge my iPhone every day just because of Norton and see if things will improve. I have attached a screenshot below:
iPhone 12 Pro Max with the latest version of iOS and Norton 360
Norton please fix your software!! 
Sorry I missed your original post.
On iOS devices I need to confirm if it is the actual VPN feature or the Internet Security > Safe Web feature you are disabling. Both will display the VPN icon in the iOS notification area at the top of the screen.
If the actual VPN feature, it would not be unusual for extra battery use as everything that is sent from and received by your device has to be encrypted/decrypted. This is going to use up additional CPU cycles and thus the battery power. If you are using cell data, you do not need the VPN as cell signals are already encrypted. You would only need the VPN when using public, unsecured wifi hotspots.
The Safe Web feature should not be too intrusive, as it would only check with Norton Safe Web servers while using a browser. And it would not need that many extra CPU cycles as not encryption is being done.
Bump. Any thing from Norton? I have noticed this issue on all the iOS and iPadOS devices!! Please optimize your software
