'tail -f' for a specific amount of time

With GNU timeout:

timeout 20 tail -f /path/to/file

For completeness sake, without timeout, you can do this:

#!/bin/sh
tail -f /var/log/syslog &
me=$!
trap "kill $me" INT TERM HUP QUIT EXIT
sleep 20

The trap line ensures that when the script or parent shell is terminated (we reach end of script (EXIT), Ctrl-C (INT), sending a SIGTERM via kill, logging out of the shell (HUP), etc) then the tail is killed.