my $out = ">./coffee.tmp";
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Does your file name have a '>' character in it? You should probably be using the three argument form of open instead.

open OUT, $out || die "Can't open $out!\n";
open FILE, $file || die "Can't open file oldcoffee!\n";
close FILE, $file || die "Can't close file oldcoffee!\n";
close OUT, $out || die "Can't close $out!\n";
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Using the high precedence || operator means that the dies are superfluous. You should use the low precedence or operator instead. close accepts one argument, the filehandle. You should also include the $! variable in the error messages so you know why the program failed.

    $mails[$#mails + 1] = $_;
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Is better written as:

    push @mails, $_;
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If you want to split the mbox file into separate files based on the year in the /^From/ line then you probably want something like this:

#!/usr/bin/perl
use warnings;
use strict;

my $file = 'oldcoffee.tmp';
my $out  = 'coffee.tmp';

open FILE, '<', $file or die "Can't open file $file! $!\n";

while ( <FILE> ) {
    if ( /^From / && /(\d{4})$/ ) {
        # Update: changed 'readonly' to 'append'
        open OUT, '>>', "$1$out" or die "Can't open $1$out! $!\n";
    }
    fileno OUT and print OUT;
}

close FILE or die "Can't close file $file! $!\n";

__END__
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