You don't need to use $_, just use $line from the first foreach and use a different variable name for the second foreach:

foreach $line (@words) {
  $line =~ tr/A-Z/a-z/;

  foreach $swear (@ignore) {
    $line =~ s/$swear//;
  }
}

But there is no need to do it nested. Just use two different loops. And I don't know about the relative merits of each, but if you want to lowercase a string, you can just use the Perl function lc.