PHP DOMDocument errors/warnings on html5-tags

No, there is no way of specifying a particular doctype to use, or to modify the requirements of the existing one.

Your best workable solution is going to be to disable error reporting with libxml_use_internal_errors:

$dom = new DOMDocument;
libxml_use_internal_errors(true);
$dom->loadHTML('...');
libxml_clear_errors();

You could also do

@$dom->loadHTML($htmlString);

You can filter the errors you get from the parser. As per other answers here, turn off error reporting to the screen, and then iterate through the errors and only show the ones you want:

libxml_use_internal_errors(TRUE);
// Do your load here
$errors = libxml_get_errors();

foreach ($errors as $error)
{
    /* @var $error LibXMLError */
}

Here is a print_r() of a single error:

LibXMLError Object
(
    [level] => 2
    [code] => 801
    [column] => 17
    [message] => Tag section invalid

    [file] => 
    [line] => 39
)

By matching on the message and/or the code, these can be filtered out quite easily.