When you do this, or attempt to do this, with Thunderbird you have to create a filter to send out a canned response. Yes, your computer has to stay on and running all the time. Thunderbird is software that runs on your computer and this requires it to stay on and connected to the Internet.

I have not seen to many email providers that do not have a way to setup a vacation response from their web mail interface. Have you checked with them? This is the way to do it and have it work.

Please check with your mail provider first; if they offer this feature then it's by far the most sensible option. They are geared up for 24/7 operation and have fallbacks and contingency plans for power outages and equipment failures, etc.

If you are going to delegate this task to Thunderbird, you must leave Thunderbird running. So you must leave your computer running, and your internet connection running. Since you may receive messages at any time, you must leave it running and connected all the time,

Why bother to send someone in to switch it off and on? (And login as you! Not safe!)

Of course, none of this necessarily avoids messages accumulating in your mailbox and filling your account, though if Thunderbird is left running, it will collect and download messages in a POP-connected account. The main benefit is that correspondents will understand that they should not expect an immediate reply from you.