Just for the fun, a HOP version using closures and function factories. First a long explanatory version:

use strict;
use warnings;

my $file1 = make_func (shift);
my $file2 = make_func (shift);

while (1) {
    my $line1 = $file1->();
    my $line2 = $file2->();
    last unless defined $line1 and defined $line2;
    print "$line1 $line2 \n";
}

sub make_func {
    my $file = shift;
    open my $FH, "<", $file or die "could not open $file $!";
    return sub 
        {my $c = <$FH>;  
        return unless defined $c; 
        chomp $c; 
        return $c
    }
}
[download]

And now a more concise form:

use strict;
use warnings;

my $file1 = make_func (shift);
my $file2 = make_func (shift);

my ($line1, $line2);
chomp $line1 and print $line1, $line2 while ($line1 = $file1->() and $
+line2 = $file2->());

sub make_func {
    my $file = shift;
    open my $FH, "<", $file or die "could not open $file $!";
    sub {<$FH>; }
}
[download]

It can probably be made more concise, but I have no time right now.