my $remove = q{<div><div class="blue"></div></div>};

Don't use quoted string constructors to make regex patterns; use  qr// (update: to make honest-to-goodness regex objects) (see perlop, perlre, perlretut, and perlrequick). Using ordinary quoted string constructors sets you up for future puzzling bugs.

my $str = q{$line};

This is a meaningless statement; it just assigns a literal  $line to a string:

c:\@Work\Perl\monks>perl -wMstrict -le
"my $str = q{$line};
 print qq{'$str'};
"
'$line'
[download]

my $str = qr{$line};

The problem here is that you seem to be trying to make the entire line you've just read from the file into a pattern. You then remove a piece of the pattern with a substitution:

c:\@Work\Perl\monks>perl -wMstrict -le
"my $remove = qr{ now \s+ brown }xms;
 ;;
 my $line = qq{how now brown cow \n};
 print qq{<$line>};
 ;;
 my $str = qr{$line};
 print $str;
 ;;
 ($line = $str) =~ s/$remove//;
 print qq{<$line>};
"
<how now brown cow
>
(?^:how now brown cow
)
<(?^:how  cow
)>
[download]

Do you see where the extraneous  (?^: ... ) stuff comes from?

Useless use of non-destructive substitution (s///r) in void context

You have to use a  s///r substitution in a statement like
    my $new_line_changed = $old_line_not_changed =~ s/$remove//gr;
(and I would recommend use of the  /g "global" modifier also).

Update: Changed variable names in last code example to (hopefully!) clarify the point being made.

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