Here's a way using index and substr:

#!/usr/bin/perl

my $str = "123.0456.0789.0-WHOOPS";
my $sub = ".0";

my $p = index($str,$sub,index($str,$sub)+1);
print "str before = $str\n";
substr($str,$p,length($sub)) = "" unless $p < 0;
print "str  after = $str\n";
[download]

Or more generally,

#!/usr/bin/perl

my $str = "123.0456.0789.0-WHOOPS";
my $sub = ".0";

print "str before = $str\n";
$str = rm_nth($str,$sub,2);
print "str  after = $str\n";

sub rm_nth {
   my ($str,$sub,$n) = @_;  $n = 1 if $n < 1;
   my $p = -1;

   while ($n--) {
      $p = index($str,$sub,$p+1);
      return $str if $p < 0;
   }
   substr($str,$p,length($sub)) = "";
   return $str;
}
[download]

I'm sure someone will post the generalized regular expression version now :-)

Although, in Perl6 it will be so much nicer to say s:2nd/$pat//; or s:nth(2)/$pat//;

Update: Slight change to the assignment of $n in rm_nth so that it handles requests for deleting negative occurrences in a non-infinite-loop sort of manner :)