Here's some data. Well, technically anecdata, I suppose.

My phone has been powered on but inactive all night; I charged it to 80% before going to bed, then unplugged it and left it where I can reach it from my bed, as is my habit. (I'm in an Asian timezone, in case you hadn't guessed, so it's morning for me while it's evening in America right now). Its battery is now at 73%. The Android battery report says 6% battery usage from Kindle (makes sense, I started reading a book when I woke up), 0.7% from Signal (haven't sent any messages yet today but have received a few), and 0.3% from Tailscale.

So when you're not using the Tailscale network actively, you'll hardly notice the battery drain.