You are right that there is unnecessary disk access going on. There are cute/clever ways to do this (e.g. piping in from grep), but the easiest way would be swap your print statement with a concatenation:

open my $FH, '<' , $config or die "Cannot open $config: $!\n";
my $data = <$FH>; # Dump the line
$data = '';
while (<$FH>) {
    next if /^\s*#|^$/;
    $data .= $_;
}
close $FH;
while ($data =~ m/\{([^}]*)\}/gx ) { 
    print "$1\n";
}
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Incidentally, you have a $data = <$FH>; peppered into your first chunk. Are you meaning to dump the first line of the file? I've included that behavior, but it smells buggy to me.