The piped open returns the pid of the child process. So just kill it once you're done with it:

#!/usr/bin/perl

$SIG{INT} = \&end;

my $kid = open (TELNET, '-|', "telnet 192.168.1.1 6088");

while(<TELNET>) {
    $arr = <TELNET>;
    $arr =~ s/[\r\n](.....):(.*)/<p class="$1">$1:$2<\/p>/;
    print $arr;
}
end;

sub end {
    print "Closing the session ...\n"
#   close FH;
    close TELNET;
    kill 2, $kid; ## Or whatever signal is appropriate!
    print "Session closed!\n"
    exit;
}
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