As you’re dealing with HTML, you should use an HTML parser to process it correctly. You can use PHP’s DOMDocument and query the elements with DOMXPath, e.g.:
$doc = new DOMDocument();
$doc->loadHTML($html);
$xpath = new DOMXPath($doc);
foreach ($xpath->query('//h1') as $node) {
$node->parentNode->removeChild($node);
}
$html = $doc->saveHTML();