Rather than iterating over the lines of "file1" each time you test a line from "file2", construct a single regex that has all the lines from "file1" as alternations.

use strict;
use warnings;


open my $file1FH, q{<}, \ <<END_OF_FILE1 or die qq{open: $!\n};
123456
123457
123458
END_OF_FILE1

chomp( my @file1Lines = <$file1FH> );

close $file1FH or die qq{close: $!\n};

my $rxFile1 = do
   {
       local $" = q{|};
       qr{^(?:@file1Lines)};
   };

open my $file2FH, q{<}, \ <<END_OF_FILE2 or die qq{open: $!\n};
123456 foo
123456 bar
123457 foobar
123455 this would not be printed to new file
END_OF_FILE2

while ( <$file2FH> )
{
    next unless m{$rxFile1};
    print;
}
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Here's the output.

123456 foo
123456 bar
123457 foobar
[download]

I hope this is of interest.

Cheers,

JohnGG