All conditionals need curly braces (such as in toHtml in HTML_QuickForm_LiveSearch_Select).
Output of HTML attributes needs to be escaped correctly (htmlentities with ENT_QUOTES). This is important for the output of values in options.
Please use single quotes in output with concatenation. (in setting $liveform you use double quotes and interpolation as well as concatenation. using only single quotes will be (very slightly) faster and easier to read).
I noticed some funky indenting in a few places, please make sure that everything is indented correctly and not using tabs.
livesearch.class.php does not fit the PEAR naming scheme. I saw this in the AJAX server class, but didn't see an actual file.
I would very much prefer the use of quotes over HEREDOC, especially in any cases where you're adding PHP variables to the strings.