How to exclude previous months of the current month?

Make the Array Keys to Ints and compare them with date('n') (n stands for month without a prefix zero)

foreach($months as $month => $monthtitle):
  if ((int)$month >= (int)date('n')){
     echo '<li title="'.$monthtitle.'" rel="'.$month.'"><a href="#" title="'.$monthtitle.'">'.$monthtitle.'</a></li>';
  }
endforeach;

You will need to check what is right now by using date() function. Specifically, the m parameter. To do that, you need to change this line:

if ($month == '2'){

into:

if ($month >= date('m')){

I think something like this can get you started:

<?php

$months = array('1' => 'January', '2' => 'February', '3' => 'March', '4' => 'April', '5' => 'May', '6' => 'June', '7' => 'July', '8' => 'August', '9' => 'September', '10' => 'October', '11' => 'November', '12' => 'December');
echo '<li><a class="by_month"></a><ul class="monthby" name="monthby">';

$a = date('n',time()); // current month (numeric)

for($i = $a; $i <= 12; $i++){
    echo '<li title="'.$months[$i].'" rel="'.$months[$i].'"><a href="#" title="'.$months[$i].'">'.$months[$i].'</a></li>';
}

echo '</ul></li>';

I would consider using the built in date function so you don't have to list the Months out manually in your code. Then I might use the mktime function to convert the months to timestamps and compare them based on that. I've put some code below and here is a link to a codepad.

$currentMonth = 2;
$currentTime = mktime(0, 0, 0, $currentMonth, 1, 2012);
for ($i = 1; $i <= 12; $i++) {
  $iTime = mktime(0, 0, 0, $i, 1, 2012);
  if ($iTime >= $currentTime) {
    $monthTitle = date('F', $iTime);
    echo '<li title="' . $monthTitle . '" rel="' . $i . '"><a href="#" title="' . $monthTitle . '">' . $monthTitle . '</a></li>';
  }
}