The “Get mail” button involves any account. There is no any option to exclude one account from the global retrieval as Outlook allows you to do, so the only way is hacking Thunderbird through the about:config editor.
In the Linux version
In Windows should be the same but starting from Tool→options→
Now we should edit the value of the key
but we don't know what's the number of the account we are interested in.
There are several ways to find that number, but I've come up with this one which is the quickest:
In the search box type server.server*.name
You'll see a list of keys with the name of the accounts in their values: pick the number for the account you want to exclude. E.g. if the key matching the account name is mail.server.server3.name the number you are interested in is 3.
Now in the search box type
where <number> is the number found previously.
You'll see only 1 row as a result. Double click it to change the value to false.
Now the get mail button won't retrive the emails for that account anymore.