wget or curl from stdin
Try piping the tail -f through python -c $'import pycurl;c=pycurl.Curl()\nwhile True: c.setopt(pycurl.URL,raw_input().strip()),c.perform()'
This gets curl (well, you probably meant the command-line curl and I'm calling it as a library from a Python one-liner, but it's still curl) to fetch each URL immediately, while still taking advantage of keeping the socket to the server open if you're requesting multiple URLs from the same server in sequence. It's not completely robust though: if one of your URLs is duff, the whole command will fail (you might want to make it a proper Python script and add try / except to handle this), and there's also the small detail that it will throw EOFError on EOF (but I'm assuming that's not important if you're using tail -f).