It looks almost paradoxical, but this does the trick:

#! perl
use strict;
use warnings;

while (<DATA>) {
  print if /(\d)(?=.*\1)/g and not /($1)(?=.*\1)/g;
}

__DATA__
0000 (Don't pick)
0001 (Don't pick)
0012 (Pick)
0011 (Pick)
1001 (Pick)
1009 (Pick)
1234 (Don't pick)
9879 (Pick)
[download]

Because I use the /g, the second match continues after the first one, so its capture group will actually be the 2nd occurrence of the digit. Then the backreference would be the third (which would mean we don't want it).

It would not work for situations where some digit appears three times but another appears only twice (so it would be a "pick"). But since your examples are only four digits long, that doesn't come up here.