How to deal with session timeouts in AJAX requests

You should only store a link to the users identity in the session. Use sessions to identify a user as x and then get user x's information from the database.

If your problem is with users sessions timing out then you should reconsider how you're using your sessions. Perhaps make them last until the browser closes? If you really want to make them a duration, then perhaps ping the server in intervals to keep the session alive.

Decide in your php script whether or not the user should be able to vote. If the session isn't set, or if they have already voted, return a message that you can identify with on the client side. If they already voted perhaps return "voted":"true" in a JSON object. Use JS to parse this object and understand what it means, taking the appropriate action. If the session isn't set, perhaps return "session_set":"false", and then make javascript redirect with a window.location = "login.php" etc.

Only increment the counter for the user on a successful return of a counted vote.

Consider returning an http status of 401, and a JSON object detailing the reason. If you're using jQuery, that'll drop you to the error() callback, which you can then parse your object.

$.ajax({
  data: {},
  dataType: 'html',
  success: function(data) {
    // do whatever here
  },
  type: 'POST',
  url: 'myserver.com',
  error: function(XMLHttpRequest, textStatus, errorThrown) {
    // XMLHttpRequest.responseText has your json string
    // XMLHttpRequest.status has the 401 status code
    if (XMLHttpRequest.status === 401) {
      location.href = 'login.php';
    }
  }
});

I'm not familiar with PHP anymore, but this should work for just about any environment. You may have to suppress any automatic login form redirection though. In asp.net mvc the framework will see the 401 and push the default login form back, with a status of 200.