Your index.php code is correct. I am including the updated code for common.php below then I will explain the differences.
<?php
$class = ($page == 'one') ? 'class="active"' : '';
$nav = <<<EOD
<div id="nav">
<ul>
<li><a $class href="index.php">Tab1</a>/</li>
<li><a href="two.php">Tab2</a></li>
<li><a href="three.php">Tab3</a></li>
</ul>
</div>
EOD;
?>
The first issue is that you need to make sure that the end declaration for your heredoc -- EOD;
-- is not indented at all. If it is indented, then you will get errors.
As for your issue with the PHP code not running within the heredoc statement, that is because you are looking at it wrong. Using a heredoc statement is not the same as closing the PHP tags. As such, you do not need to try reopening them. That will do nothing for you. The way the heredoc syntax works is that everything between the opening and closing is displayed exactly as written with the exception of variables. Those are replaced with the associated value. I removed your logic from the heredoc and used a tertiary function to determine the class to make this easier to see (though I don't believe any logical statements will work within the heredoc anyway)
To understand the heredoc syntax, it is the same as including it within double quotes ("), but without the need for escaping. So your code could also be written like this:
<?php
$class = ($page == 'one') ? 'class="active"' : '';
$nav = "<div id=\"nav\">
<ul>
<li><a $class href=\"index.php\">Tab1</a>/</li>
<li><a href=\"two.php\">Tab2</a></li>
<li><a href=\"three.php\">Tab3</a></li>
</ul>
</div>";
?>
It will do exactly the same thing, just is written somewhat differently. Another difference between heredoc and the string is that you can escape out of the string in the middle where you can't in the heredoc. Using this logic, you can produce the following code:
<?php
$nav = "<div id=\"nav\">
<ul>
<li><a ".(($page == 'one') ? 'class="active"' : '')." href=\"index.php\">Tab1</a>/</li>
<li><a href=\"two.php\">Tab2</a></li>
<li><a href=\"three.php\">Tab3</a></li>
</ul>
</div>";
?>
Then you can include the logic directly in the string like you originally intended.
Whichever method you choose makes very little (if any) difference in the performance of the script. It mostly boils down to preference. Either way, you need to make sure you understand how each works.