copying your code from PM works for me, but im pretty sure you have a unicode issue

compare

use strict;
use warnings;
use Data::Dump qw/pp dd/;
#use utf8;


my $string = 'copied from a web page looking like this: “Test.” ';

dd $string;

$string =~ s/“//g;
$string =~ s/”//g;

dd $string;
[download]

without utf8 strings are an octect-stream

C:/Strawberry/perl/bin\perl.exe -w d:/tmp/pm/replace_quotes.pl 
"copied from a web page looking like this: \xE2\x80\x9CTest.\xE2\x80\x
+9D "
"copied from a web page looking like this: Test. "
[download]

with utf8 they are in the internal unicode variant

C:/Strawberry/perl/bin\perl.exe -w d:/tmp/pm/replace_quotes.pl 
"copied from a web page looking like this: \x{201C}Test.\x{201D} "
"copied from a web page looking like this: Test. "
[download]

you need to make sure that the chars in your file (i.e. the way it's saved by your editor) and the web-input are both in the same encoding, because “” are not ASCII-characters.

see perlunitut