Add or die(mysql_error()); to the end to display any errors. Try again and post the error you receive.
I also added ECHO to the variable assignment. Make sure you are actually submitting data.
include "db_connect.php";
echo $update = $_POST['section1'];
mysql_query("UPDATE `sections` SET section1='$update'")or die(mysql_error());
Also, you need to give the query a WHERE statement. What other columns do you have in your db?
btw, if you are making a public script, using posted data to manipulate a db is very unsafe. If you haven't already, look into SQL Injection.