Others have provided alternatives to looking at your watch. The coding, if inelegant, is straightforward:

#!/usr/bin/perl -lw
use strict;

my $open = 0;
my $hour = 0;

my $x = localtime; 
# $x was "Wed Jan 14 21:59:45 2009" as this was written

if ( $x=~ /(Mon|Tue|Wed|Thu|Fri)\s\w{3}\s\d{1,2}\s(\d\d):/ ) {
    my $weekday = $1; 
    $hour = $2;
}

if ( (7 <= $hour) && ($hour < 17) ) {
    my $open = 1;
    print "Open is true";           # debug only
    # print <img src = "open.gif" ...;
} else {
    #print <img src = "closed.gif ...;
    print "Live chat available 7am - 5pm (appropriate TZ)";
}
[download]

integrating with CGI is left as an exercise...

Afterthought: Of course (duh!) you'll also have to test for holidays which might be accomplished by testing (with another regex) localtime against a hash of holiday_name = holiday_date, updated each year.

Update: Fixed day names.