$ cat test.txt
<h3>search engine links</h3>
<a href="http://google.com">google</a>
<a href="http://www.bing.com">bing</a>
<a href="http://www.yahoo.com">yahoo</a>
<h3>payment links</h3>
<a href="http://www.paypal.com">paypal</a>
<a href="http://www.paxum.com">paxum</a>
$ sed -e '/<h3>payment/,/<\/h3>/ d' -e '/<h3>/ d' test.txt
<a href="http://google.com">google</a>
<a href="http://www.bing.com">bing</a>
<a href="http://www.yahoo.com">yahoo</a>
$