The http protocol does not directly support what you're trying to do. It's designed to return an entire file specified in an http request, not parts of a file. The easiest solution would be to slurp a copy of the file into a variable. Here are a couple of ways using LWP::Simple.

use LWP::Simple;

# get file into variable
my $data = get('http://www.jaywil.com/libsearch/SAMPLE.TXT');

# print all at once
print $data;

# or break up and print line by line
 # update: to keep blank lines at eof see comment by ikegami below 
@lines = split(/\n/, $data);
for (@lines){
    print "$_\n";
}
[download]

If for some reason you actually need to get the file one line at a time over http, you would need some help on the server side. If someone hasn't already written a module for this, you could write a cgi script that keeps state information and handles a few operations such as *open*, *read_line* and *close*. Basically you would be designing your own mini protocol to run over http.