I would code it like this (no need for the print before the loop and also swap order of statements within the loop):

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

my @array=(1..10);

while (@array)
{
   print "@array\n";
   shift (@array);   
}

__END__

1 2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10
3 4 5 6 7 8 9 10
4 5 6 7 8 9 10
5 6 7 8 9 10
6 7 8 9 10
7 8 9 10
8 9 10
9 10
10
[download]

A for() or foreach() loop calculates some stuff before it starts looping that is not "re-calculated". That is both "good" and "bad". The while(cndx){} construct recalculates the cndx at each iteration.

Update: This may confuse more than it helps, but an attempt...
When you iterate over foreach(@array), the default variable $_ is being set to each element of @array. The shift mucks around with this progression.

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

my @array=(1..10);
print "@array\n";

foreach (@array)
{
  #shift (@array);  #see below
  print "default var = $_  @array\n";
}

__END__

1 2 3 4 5 6 7 8 9 10
default var = 1  1 2 3 4 5 6 7 8 9 10
default var = 2  1 2 3 4 5 6 7 8 9 10
default var = 3  1 2 3 4 5 6 7 8 9 10
default var = 4  1 2 3 4 5 6 7 8 9 10
default var = 5  1 2 3 4 5 6 7 8 9 10
default var = 6  1 2 3 4 5 6 7 8 9 10
default var = 7  1 2 3 4 5 6 7 8 9 10
default var = 8  1 2 3 4 5 6 7 8 9 10
default var = 9  1 2 3 4 5 6 7 8 9 10
default var = 10  1 2 3 4 5 6 7 8 9 10

With the "shift turned on" (not commented out)
1 2 3 4 5 6 7 8 9 10
default var = 1  2 3 4 5 6 7 8 9 10
default var = 3  3 4 5 6 7 8 9 10
default var = 5  4 5 6 7 8 9 10
default var = 7  5 6 7 8 9 10
default var = 9  6 7 8 9 10
[download]