Lumix tz8 won't turn on

Are you saying that there's no electrical power at all? Or do you mean that the camera won't complete its startup sequence, and gives you an error? I'll assume the second first, as that is more common:Lens errors are fairly common. Usually it's sand or grit interfering with the lens extension mechanism. Or the camera's been dropped with the lens extended. Or the camera has been powered on, but the lens had been blocked preventing its extension. Or the battery ran down with the lens extended ...
Here are some troubleshooting steps that you can try that may (or may not) correct it. They only seem to work for less than 40% of the lens errors, but if the camera is out of warranty (or repair cost approaches that of the camera), they're worth that try. Some of the later steps do involve some risk to the camera, so carefully weigh your options before deciding to conduct them.
And here's a 16 minute video summary of the info shown in the link. But note that the article link above goes into even greater detail than this video:
You-Tube video "AkEfnF6zVxc" placeholder (do not delete)



But if you really meant that the camera is not showing any electrical power at all, then see this.