You'll need to remove the header() if you want to do that. As it is now, it will redirect to location. You don't need to do anything else since there is an include() call (which although is probably not what you want, it is sufficient since the text is plain/html). It needs some work to deal with isset()'s properly in order to prevent errors, but it would run otherwise.
These branches confuse me. There seems to be no direct relationship to the if/elseif combinations for them. The flow seems chopped up unnecessarily.
Personally I don't like the format of the stored data. Assuming that every record within contains the same format, it would be more efficient to simply store, line by line, $website, $name, $email, $time and $message, and parse when reading. Simple example of that using fgetcsv (assuming you csv delimit them):
if ($fh = fopen('comments.txt', 'r+'))
while (false !== ($line = fgetcsv($fh)))
list($website, $name, $email, $time, $message) = $line;
printf('<h4>Hi, <a href="http://%s" target="_blank">%s</a>. <a href="mailto://%s" target="_blank">%s</a></h4><br/><h2>%s</h2>' . PHP_EOL,
date('F j Y \a\t h:i:s', $time),
Simple as that. That saves tons in space and is a little less rigid.