For a start you have to create a new instance of the file. Perl helps by facilitating inplace editing of files using @ARGV, $^I and while (<>) {...} magic.

The other less common element Perl used in this code is the scalar context range operator ...

use strict;
use warnings;

open OUTFILE, '>', 'delme.txt';
print OUTFILE <<TEXT;
zone "dontchangethisdomain.com" {
type master;
file "/somepath/to/dontchangethisdomain.com";
notify yes;
}
zone "mydomain.com" {
type master;
file "/path/to/zone/mydomain.com";
notify yes;
}
zone "leavethisalonetoo.com" {
type master;
file "/other/path/to/leavethisalonetoo.com";
notify yes;
}
TEXT
close OUTFILE;

my $targetdomain = 'mydomain.com';

@ARGV = ('delme.txt');
$^I = '.bak';

while (<>) {
    print "#" if m/^zone "\Q$targetdomain\E" {/ .. m/^}/;
    print;
}
[download]

Generates delme.txt containing:

zone "dontchangethisdomain.com" {
type master;
file "/somepath/to/dontchangethisdomain.com";
notify yes;
}
#zone "mydomain.com" {
#type master;
#file "/path/to/zone/mydomain.com";
#notify yes;
#}
zone "leavethisalonetoo.com" {
type master;
file "/other/path/to/leavethisalonetoo.com";
notify yes;
}
[download]