Going with a typical Apache access log, you can run:
tail -f access_log | grep \"127.0.0.1\"
Which will only show you the logs (as they are
This is the result of buffering, it will eventually print when enough data is available.
Use the --line-buffered option as suggested by Shawn Chin or if stdbuf is available you can get the same effect with:
--line-buffered
stdbuf
tail -f access_log | stdbuf -oL grep "127.0.0.1" | grep -v ".css"