You need to add the encoding used in the document at the start of it: <?xml version="1.0" encoding="ISO-8859-1" standalone="yes"?>. The entities are then converted to the proper numerical entities.

You may also want to use mirod's XML::Twig, which, in the latest version (3.0) is able to keep the original encoding. And besides this cool feature, I find Twig easier to use than DOM to process XML docs. Here's what your code would look like with Twig:

#!/usr/bin/perl -w
use strict;
use XML::Twig 3.0;

my $parser = new XML::Twig( keep_encoding => 1 );
my $xmlstring=<<"XMLEND";
<?xml version="1.0" encoding="ISO-8859-1"?>
<ACTION>
    <INPUT LABEL="Radio Button"/>
    <INPUT LABEL="été"/>
    <RADIO ID="List">
        éééàààùùù
    </RADIO>
</ACTION>
XMLEND
$parser->parse($xmlstring);
$parser->print;
[download]

Hope this helps!

update: version 3.0 of XML::Twig can be found here