Two ideas:

a) You can do this by implementing some javascript, which switches display status of each ticket after selection changed.

if ($_SESSION['staff'] >='6')
{

    echo "<select name='type' id='type' onchange='typeChanged()' onpropertychange='typeChanged()'>
    <option value='tab1'>Open Tickets</option>
    <option value='tab2'>New Tickets</option>
    <option value='tab3'>Under Review</option>
    <option value='tab4'>Closed Tickets</option>
    </select><BR>"; 
    echo "Current listed support tickets: <BR><BR>";

    echo "<div id="tab1" name="tabs">";
            $result1 = mysql_query("SELECT * FROM support WHERE status = 'New Ticket' OR 'Under Review'");
            while ($row1 = mysql_fetch_array($result1))
            {
                $ticket1 = $row1['TicketID'];
                $user1 = $row1['name'];
                echo "Ticket ID: " . $row1['TicketID'] . "<BR>" . $row1['type'] . "<BR>Subject: " . $row1['subject'] . "<BR>Ticket Status: " . $row1['status'] . "<BR>";
                echo "<form action='detail_ticket.php?t=$ticket1' method='post' name='detail_t' id='detail_t'><input type='submit' name='detail_b' id='detail_b' value='View Ticket!'></form>";
                echo "<form action='reply_ticket.php?r=$user1' method='post' name='r_ticket' id='r_ticket'><input type='submit' name='reply_b' id='reply_b' value='Reply on Ticket!'></form>";
            }
    echo "</div>";

    echo "<div id="tab2" name="tabs">";
            $result2 = mysql_query("SELECT * FROM support WHERE status = 'New Ticket'");
            while ($row2 = mysql_fetch_array($result2))
            {
                $ticket2 = $row2['TicketID'];
                $user2 = $row2['name'];
                echo "Ticket ID: " . $row2['TicketID'] . "<BR>" . $row2['type'] . "<BR>Subject: " . $row2['subject'] . "<BR>Ticket Status: " . $row2['status'] . "<BR>";
                echo "<form action='detail_ticket.php?t=$ticket2' method='post' name='detail_t' id='detail_t'><input type='submit' name='detail_b' id='detail_b' value='View Ticket!'></form>";
                echo "<form action='reply_ticket.php?r=$user2' method='post' name='r_ticket' id='r_ticket'><input type='submit' name='reply_b' id='reply_b' value='Reply on Ticket!'></form>";
            }
    echo "</div>";

    echo "<div id="tab3" name="tabs">";
            $result3 = mysql_query("SELECT * FROM support WHERE status = 'New Ticket'");
            while ($row3 = mysql_fetch_array($result3))
            {
                $ticket3 = $row3['TicketID'];
                $user3 = $row3['name'];
                echo "Ticket ID: " . $row3['TicketID'] . "<BR>" . $row3['type'] . "<BR>Subject: " . $row3['subject'] . "<BR>Ticket Status: " . $row3['status'] . "<BR>";
                echo "<form action='detail_ticket.php?t=$ticket3' method='post' name='detail_t' id='detail_t'><input type='submit' name='detail_b' id='detail_b' value='View Ticket!'></form>";
                echo "<form action='reply_ticket.php?r=$user3' method='post' name='r_ticket' id='r_ticket'><input type='submit' name='reply_b' id='reply_b' value='Reply on Ticket!'></form>";
            }
    echo "</div>";

    echo "<div id="tab4" name="tabs">";
            $result4 = mysql_query("SELECT * FROM support WHERE status = 'Closed'");
            while ($row4 = mysql_fetch_array($result4))
            {
                $ticket4 = $row4['TicketID'];
                $user4 = $row4['name'];
                echo "Ticket ID: " . $row4['TicketID'] . "<BR>" . $row4['type'] . "<BR>Subject: " . $row4['subject'] . "<BR>Ticket Status: " . $row4['status'] . "<BR>";
                echo "<form action='detail_ticket.php?t=$ticket4' method='post' name='detail_t' id='detail_t'><input type='submit' name='detail_b' id='detail_b' value='View Ticket!'></form>";
                echo "<form action='reply_ticket.php?r=$user4' method='post' name='r_ticket' id='r_ticket'><input type='submit' name='reply_b' id='reply_b' value='Reply on Ticket!'></form>";
            }
    echo "</div>";


    echo "Click <A href='overview_page.php'>HERE</A> to return to the overview page.<BR>";
    echo "Click <A href='uitlogen_user.php'>HERE</A> to logout!<BR>";
}else
{
    echo "You need to be an GM or Admin to view this page!<BR>";
    echo "Click <A href='uitlogen_user.php'>HERE</A> to login!";
}

?>

<script type="text/javascript">
function typeChanged()
{
    var chosen = document.getElementById('type').value;
    var tabs = document.getElementsByName('tabs');
    for (var i in tabs)
    {
        tabs[i].style.display = 'none';
    }
    document.getElementById(chosen).style.display = 'block';
}
</script>

b) Redirect to this very page with some url parameter indiciates which ticket is selected, and display the corresponding one.

<?PHP
session_start();
include('connection.php');

if ($_SESSION['staff'] >='6')
{
    $chosen = isset($_GET['chosen']) ? $_GET['chosen'] : 'tab1';

    echo "<select name='type' id='type' onchange='typeChanged()' onpropertychange='typeChanged()'>
    <option value='tab1'>Open Tickets</option>
    <option value='tab2'>New Tickets</option>
    <option value='tab3'>Under Review</option>
    <option value='tab4'>Closed Tickets</option>
    </select><BR>"; 
    echo "Current listed support tickets: <BR><BR>";

    if ($chosen == 'tab1')
    {
            $result1 = mysql_query("SELECT * FROM support WHERE status = 'New Ticket' OR 'Under Review'");
            while ($row1 = mysql_fetch_array($result1))
            {
                $ticket1 = $row1['TicketID'];
                $user1 = $row1['name'];
                echo "Ticket ID: " . $row1['TicketID'] . "<BR>" . $row1['type'] . "<BR>Subject: " . $row1['subject'] . "<BR>Ticket Status: " . $row1['status'] . "<BR>";
                echo "<form action='detail_ticket.php?t=$ticket1' method='post' name='detail_t' id='detail_t'><input type='submit' name='detail_b' id='detail_b' value='View Ticket!'></form>";
                echo "<form action='reply_ticket.php?r=$user1' method='post' name='r_ticket' id='r_ticket'><input type='submit' name='reply_b' id='reply_b' value='Reply on Ticket!'></form>";
            }
    }

    if ($chosen == 'tab2')
    {
            $result2 = mysql_query("SELECT * FROM support WHERE status = 'New Ticket'");
            while ($row2 = mysql_fetch_array($result2))
            {
                $ticket2 = $row2['TicketID'];
                $user2 = $row2['name'];
                echo "Ticket ID: " . $row2['TicketID'] . "<BR>" . $row2['type'] . "<BR>Subject: " . $row2['subject'] . "<BR>Ticket Status: " . $row2['status'] . "<BR>";
                echo "<form action='detail_ticket.php?t=$ticket2' method='post' name='detail_t' id='detail_t'><input type='submit' name='detail_b' id='detail_b' value='View Ticket!'></form>";
                echo "<form action='reply_ticket.php?r=$user2' method='post' name='r_ticket' id='r_ticket'><input type='submit' name='reply_b' id='reply_b' value='Reply on Ticket!'></form>";
            }
    }

    if ($chosen == 'tab3')
    {
            $result3 = mysql_query("SELECT * FROM support WHERE status = 'New Ticket'");
            while ($row3 = mysql_fetch_array($result3))
            {
                $ticket3 = $row3['TicketID'];
                $user3 = $row3['name'];
                echo "Ticket ID: " . $row3['TicketID'] . "<BR>" . $row3['type'] . "<BR>Subject: " . $row3['subject'] . "<BR>Ticket Status: " . $row3['status'] . "<BR>";
                echo "<form action='detail_ticket.php?t=$ticket3' method='post' name='detail_t' id='detail_t'><input type='submit' name='detail_b' id='detail_b' value='View Ticket!'></form>";
                echo "<form action='reply_ticket.php?r=$user3' method='post' name='r_ticket' id='r_ticket'><input type='submit' name='reply_b' id='reply_b' value='Reply on Ticket!'></form>";
            }
    }

    if ($chosen == 'tab4')
    {
            $result4 = mysql_query("SELECT * FROM support WHERE status = 'Closed'");
            while ($row4 = mysql_fetch_array($result4))
            {
                $ticket4 = $row4['TicketID'];
                $user4 = $row4['name'];
                echo "Ticket ID: " . $row4['TicketID'] . "<BR>" . $row4['type'] . "<BR>Subject: " . $row4['subject'] . "<BR>Ticket Status: " . $row4['status'] . "<BR>";
                echo "<form action='detail_ticket.php?t=$ticket4' method='post' name='detail_t' id='detail_t'><input type='submit' name='detail_b' id='detail_b' value='View Ticket!'></form>";
                echo "<form action='reply_ticket.php?r=$user4' method='post' name='r_ticket' id='r_ticket'><input type='submit' name='reply_b' id='reply_b' value='Reply on Ticket!'></form>";
            }
    }


    echo "Click <A href='overview_page.php'>HERE</A> to return to the overview page.<BR>";
    echo "Click <A href='uitlogen_user.php'>HERE</A> to logout!<BR>";
}else
{
    echo "You need to be an GM or Admin to view this page!<BR>";
    echo "Click <A href='uitlogen_user.php'>HERE</A> to login!";
}

?>

<script type="text/javascript">
function typeChanged()
{
    var chosen = document.getElementById('type').value;
    window.location.href = 'FILENAME.php?chosen=' + chosen;
}
</script>