I want to run a few tasks in shell.
tail -f debug|tee -a test.log
My question is: how to make the command tail -f debug|tee -a test.log
run in background, so that I can run other tasks then in shell script?
You don't need tee at all for this, just use the shell's built-in append operator:
tail -f debug >> test.log &
The trailing & works as normal in the shell. You only need tee to send the output to a file as well as standard out, which if you have it in the background probably isn't what you want.