Yes - a small queue would do this. Since you're so new, you'll do yourself a world of good by picking up a copy of the inexpensive book Learning Perl (also known as the Llama book) from O'Reilly.

while (<>)
{
    push @queue, $_;
    print shift @queue if @queue > 2;
}
[download]

#!/usr/bin/perl

use strict;
use warnings;

use Tie::File;

my $file = "file.txt";
my @lines;

tie @lines, 'Tie::File', $file or die $!;

print $_,"\n" for (@lines[0 .. $#lines-2]);

untie @lines;
[download]

$lines = `/usr/bin/wc -l afile.txt`;
open(F, '<afile.txt') || die 'cannot open afile.txt';
$cntr = 0;
while (<F>) {
   $cntr++;
    print;
    last if $cntr == $lines - 2;
}
[download]

There is a magic variable that keeps track of the current line number from the last-read filehandle. This would make your while loop a bit simpler:

  $. < $lines - 2 and print while <F>
[download]

open(F, '+<afile.txt') || die 'cannot open afile.txt';
while (<F>) {
    push(@addrs, tell(F)) unless eof(F);
}
truncate(F, $addrs[-2]) || die 'cannot truncate afile.txt';
[download]

                  1         2         3         4          
         1234567890123456789012345678901234567890123456789
perl -e 'open(F,'f');@f=<F>;pop@f;pop@f;print"$_"foreach@f'
[download]

You can remove the pops and change the print to print@f[0..$#f-2] for a savings of 13 strokes.

----
send money to your kernel via the boot loader.. This and more wisdom available from Markov Hardburn.

Why a counter and imply it's short when you have a foreach, and a print"$_" ?

print for@f[0..-2] is shorter.

Edit: Wow, I really do live in a little fluffy bunny world where everyone's happy, there's no war, pizza isn't fattening, Darth Vader was never an annoying child, the Six million Dollar Man really was a good tv show, new shoes are comfy, my girlfriend's bum never looks big in this, and code like this works.

Kind of surprised I didn't lose all the xp I'd ever gotten on this one. hardburn posted the correct code below.