One of my HA installations is running on a site with a limited monthly internet data quota. I find that, in periodes (for months) when I am not present at this site (and therefore have a reduced data quota), HA is “eating” too much internet access. I know I can restrict this in the router, but that is not a satisfactory solution, because I need to be able to access the site remotely via Tailscale. However, I do not need update checks and other internet usage during these periods.
I want a solution that enables me to swich unnecessary internet access on and off from inside the HA GUI. Is this possible?
Sure, just disable your cloud integrations. If you want to do that with an automation rather than manually then check out the Device actions the Spook integration provides.
As Tom says above, And AO is also to know your Devices, Integration, etc, If You Don’t You most likely have tons of entities running for no real purpose, talking to it’s Integrations which pass on information to it’s cloud-integrations etc.
I used an LTE-Router for about 11Month, with a 10 or 20 GB Subscription ( cant remember, but i think it was 10 ), anyhow during that period, i also “saved-up” +100 GB.
i.e
Disable all Autoupdates/Updates entities ( i do even for a no limit fiber-connection anyways )
If You have HA running in a VM, Block the i.e Windows/Linux host from internet access, As it’s only suppose to “serve” the installed VM-Manager, nothing else.
Disable Integrations/Services/Devices/Etc. which you have no use for, while away , This can be done with i.e An Away Scene or automation as Tom suggests.
" Eating , too much " Sounds Alot, as you also describes it, but what is it caused by which Devices/Entities/Integrations/etc ? … Monitor/enable logging in Your Router, for a period, and you’ll see which External Url’s is requested , from HA, this will help you, analyzing your system’s behavior. ( Btw What is " Eating to much", in your case ? )
I find it hard to believe HA have any reason to use near i.e 10GB Data-Traffic in a month.
But yes Don’t “restrict” remote-access to your Device (HA ) that’s obvious ! ( Not sure why you mention that at all, as that’s Not what you Need to Limit )
So Monitor/Log on your Router, so you find the Culprit, i.e is it outgoing or incoming traffic, from where to where etc ?
( You Might Find that You are the “cause”, by accessing your remote site, constantly streaming high-res cams etc 
)
PS. And don’t have an i.e APP or similar where you stream your i.e Cams in High-res. continuously to a Device-Brand-Services etc, even while you don’t access it
( certain Cams might also be a “point” where you could block/limit internet access ( you can still view it in HA , via your Tailscale-Connection)