Consider:

use strict;
use warnings;

my @insert1 =  qw(1 1 1);
my @insert2 =  qw(2 2 2);
my @inserts = (\@insert1, \@insert2);

my $outfile = 'delme.txt';
my $ins = 0;

open OUTFILE, '>', $outfile;
print OUTFILE <<DATA;
first second third
1 2 3
x y z
DATA
close OUTFILE;

for (@inserts) {
    insert (@$_);
    local @ARGV = $outfile;
    local $/ = undef;
    print <>;
}

...
[download]

Prints:

insert called:0
first    second    third    1    
1    2    3    1    
x    y    z    1    
insert called:1
first    second    third    1    2    
1    2    3    1    2    
x    y    z    1    2
[download]

Note that the bogus result from your reworking of my code was due to $/ being set to undef and therefore the <FILE2> seeing the entire file as one line on the second iteration. Sometimes I include "interesting" ways of doing stuff to make you explore the language and the documentation a little. In this case read perlvar to gain a little insight. ;)