As long as you can guarantee that the substring to match will not extend over more than two lines, you could use a two-line sliding buffer that you apply the substitution to, but then split again before outputting (so you won't get everything twice):

#!/usr/bin/perl
use strict;
use warnings;

my $prev = "";
my $done = 0;

while (<DATA>) {

    # sliding two-line buffer
    $_ = $prev.$_;

    # get rid of middle newline, so it doesn't interfere with match
    s/\n//;

    s/<\?string\?>/<xxx>/g;

    # split buf
    # first part will be output, second part kept
    # for joining with next input line
    ($_, $prev) = unpack "a".length($prev)."a*", $_;

    print "$_\n" if 2 .. $done;

    # corner case: have last line be printed as well
    if (eof and !$done++) { $_ = ""; redo }
}

__DATA__
hello there this is a bad <?string?> that 
we need to take away. Here is another bad <?st
ring?> that needs to go too.
[download]

Output:

hello there this is a bad <xxx> that 
we need to take away. Here is another bad <xxx
> that needs to go too.
[download]

As you can see, line breaks may be occur anywhere in the middle of the substitution. If that's a problem, you'd have to modify the logic of splitting the sliding buffer (which can get tricky...).

(There may be corner cases I've overlooked, but you get idea...)