Calling a save function from a “submit” button

You need to have a <form> element. By not specifying a form action attribute the form posts to the same page you are on which makes this easy to process.

Also an id on the textarea is not enough you need to add a name attribute to catch with $_POST for processing. Also adding a nonce field is good practice, and checking permissions to be safe. Try something like this:

 function blank_overview() {

     $logPath = dirname(__FILE__).'/devlog.txt';
     $logText = file_get_contents($logPath);

     if (isset($_POST['devlog_content'])) {
         if ( (wp_verify_nonce($_POST['devlog_nonce_field'],'devlog_nonce_action'))
           && (current_user_can('manage_options')) ) {
              $logText = $_POST['devlog_content'];
              file_put_contents($logPath, $logText);
              echo "Dev Log Saved.<br><br>";
          } else {echo "Warning: Dev Log NOT Saved.<br><br>";}
     }
?>

<style>#devlog-content {width: 100%;}</style>
<div class="wrap"><h2>Labyrith Development Log</h2>
<p>A simple overview of our work.</p>
<form method="post">
<?php wp_nonce_field('devlog_nonce_action','devlog_nonce_field'); ?>
<textarea name="devlog_content" id="devlog-content"><?php echo $logText ?></textarea>
<input type="submit" class="button-primary" value="Save Dev Log">
</form>
</div>
<?php } ?>

Also you will not I have removed the exit. It is not needed here and can prevent admin footer scripts and suchlike from completing because it stops further output. The function will automatically return and allow other things to complete without it.