Why does get_the_time(‘F j’) return November 30 for all posts?

The function get_the_time is only designed to work either in the main WordPress loop or a custom loop using WP_Query, so the way you are using it won’t work correctly. The following is the right way to use it, using a custom WP_Query loop:

<?php
// Get years that have posts
$years = $wpdb->get_results( "SELECT DISTINCT YEAR(post_date) AS year FROM $wpdb->posts WHERE post_type="post" AND post_status="publish" ORDER BY year ASC;" );

//  For each year, do the following
foreach ( $years as $year ) {    
    // Get all posts for the year.  Using WP_Query instead of custom mySQL
    $posts_this_year = new WP_Query( array(
        'post_type' => 'post',
        'post_status' => 'publish',
        'year' => $year->year
    ) );    

    // Display the year as a header
    echo '<h1>' . $year->year . '</h1>';

    // Start an unorder list
    echo '<ul class="posts-in-year">';

    // As long as you have posts for that year, do the following.  Using the Loop.  get_the_date now works
    while ( $posts_this_year->have_posts() ) {
        //Makes current post available to template tag functions like get_the_time
        $posts_this_year->the_post(); 
        // Display the title as a hyperlinked list item
        echo '<li><a href="' . get_permalink() . '">' . get_the_title() . '</a>' . ' ' . '&mdash;' . ' ' .  get_the_time('F j') . '</li>';
    }

    //Reset post data.  Important to do this so not to mess with main loop
    wp_reset_postdata();

    // End the unordered list
    echo '</ul>';
}

?>

The Loop, which I was referring to earlier, is when you use WP_Query to iterate through the posts for you. When you do it that way, functions like get_the_time work properly because they can pull the info for the current post in the loop. For more information, read The codex data on WP_Query and multiple loops