Eliminar atributo de estilo de las etiquetas HTML

Algo como esto debería funcionar (advertencia de código no probado):

<?php

$html = '<p style="asd">qwe</p><br /><p class="qwe">qweqweqwe</p>';

$domd = new DOMDocument();
libxml_use_internal_errors(true);
$domd->loadHTML($html);
libxml_use_internal_errors(false);

$domx = new DOMXPath($domd);
$items = $domx->query("//p[@style]");

foreach($items as $item) {
  $item->removeAttribute("style");
}

echo $domd->saveHTML();

Comenté la función de @Mayerln. Funciona, pero DOMDocument realmente aporta codificación. Aquí está mi versión simplehtmldom

function stripAttributes($html,$attribs) {
    $dom = new simple_html_dom();
    $dom->load($html);
    foreach($attribs as $attrib)
        foreach($dom->find("*[$attrib]") as $e)
            $e->$attrib = null; 
    $dom->load($dom->save());
    return $dom->save();
}

Aquí tienes:

<?php

$html = '<p style="border: 1px solid red;">Test</p>';
echo preg_replace('/<p style="(.+?)">(.+?)<\/p>/i', "<p>$2</p>", $html);

?>

Por cierto, como han señalado otros, no se sugieren expresiones regulares para esto.

Yo uso esto:

function strip_word_html($text, $allowed_tags = '<a><ul><li><b><i><sup><sub><em><strong><u><br><br/><br /><p><h2><h3><h4><h5><h6>')
{
    mb_regex_encoding('UTF-8');
    //replace MS special characters first
    $search = array('/&lsquo;/u', '/&rsquo;/u', '/&ldquo;/u', '/&rdquo;/u', '/&mdash;/u');
    $replace = array('\'', '\'', '"', '"', '-');
    $text = preg_replace($search, $replace, $text);
    //make sure _all_ html entities are converted to the plain ascii equivalents - it appears
    //in some MS headers, some html entities are encoded and some aren't
    //$text = html_entity_decode($text, ENT_QUOTES, 'UTF-8');
    //try to strip out any C style comments first, since these, embedded in html comments, seem to
    //prevent strip_tags from removing html comments (MS Word introduced combination)
    if(mb_stripos($text, '/*') !== FALSE){
        $text = mb_eregi_replace('#/\*.*?\*/#s', '', $text, 'm');
    }
    //introduce a space into any arithmetic expressions that could be caught by strip_tags so that they won't be
    //'<1' becomes '< 1'(note: somewhat application specific)
    $text = preg_replace(array('/<([0-9]+)/'), array('< $1'), $text);
    $text = strip_tags($text, $allowed_tags);
    //eliminate extraneous whitespace from start and end of line, or anywhere there are two or more spaces, convert it to one
    $text = preg_replace(array('/^\s\s+/', '/\s\s+$/', '/\s\s+/u'), array('', '', ' '), $text);
    //strip out inline css and simplify style tags
    $search = array('#<(strong|b)[^>]*>(.*?)</(strong|b)>#isu', '#<(em|i)[^>]*>(.*?)</(em|i)>#isu', '#<u[^>]*>(.*?)</u>#isu');
    $replace = array('<b>$2</b>', '<i>$2</i>', '<u>$1</u>');
    $text = preg_replace($search, $replace, $text);
    //on some of the ?newer MS Word exports, where you get conditionals of the form 'if gte mso 9', etc., it appears
    //that whatever is in one of the html comments prevents strip_tags from eradicating the html comment that contains
    //some MS Style Definitions - this last bit gets rid of any leftover comments */
    $num_matches = preg_match_all("/\<!--/u", $text, $matches);
    if($num_matches){
        $text = preg_replace('/\<!--(.)*--\>/isu', '', $text);
    }
    $text = preg_replace('/(<[^>]+) style=".*?"/i', '$1', $text);
return $text;
}

Además de la respuesta de Lorenzo Marcon:

Usando preg_replace para seleccionar todo excepto el atributo style:

$html = preg_replace('/(<p.+?)style=".+?"(>.+?)/i', "$1$2", $html);

Podría manejarlo del lado del cliente, lo más fácil sería con jQuery. Algo como:

$("#tinyMce p").removeAttr("style");