First pass at it...

#!/usr/bin/perl

use strict; # https://www.perlmonks.org/?node_id=11154597
use warnings;

my %uniq;
my $n = 0;
while( <DATA> )
  {
  $uniq{ $_ }++;
  $n++;
  }
grep { $n + 1 < $_ * 2 } values %uniq and die "Cannot be done!\n";
my @norep;
while( %uniq )
  {
  my @allowed = grep { @norep == 0 || $_ ne $norep[-1] } keys %uniq;
  my @choices = grep { $uniq{$_} * 2 >= $n + 1 } @allowed; # required 
+choices
  @choices or @choices = @allowed or die "FAILED at n = $n\n", @norep;
  @choices = map { ($_) x $uniq{$_} } @choices; # weighting choices
  push @norep, my $pick = $choices[ rand @choices ];
  --$uniq{ $pick } or delete $uniq{ $pick };
  $n--;
  }
print @norep;

__DATA__
Line 1
Line 2
Line 3
Line 3
Line 3
Line 4
Line 8
[download]

Sample Output:

Line 3
Line 2
Line 1
Line 3
Line 8
Line 4
Line 3
[download]

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