Wordpress - How to remove these fields from the 'Profile' section?

For the contact methods filter: user_contactmethods:

function update_contact_methods( $contactmethods ) {

    unset( $contactmethods['aim'] );
    unset( $contactmethods['jabber'] );
    unset( $contactmethods['yim'] );

    return $contactmethods;

}
add_filter( 'user_contactmethods', 'update_contact_methods' );

Though the user_contactmethods filter is the one of the few for the user profile screen, since the other fields are not required in WordPress, you could use jQuery's .hide() and .remove() to effectively get rid of the ones you don't want without complications in terms of saving the information. Sure, a person without JS on would still see them, but that's a small minority and I'm guessing that the information changing there won't actually affect anything.

An example of removing the whole personal options piece (show admin bar and color scheme). I know it's not what you're looking for exactly, just wanted to give you a quick example I have on hand:

// remove personal options block
if( is_admin() ){
    remove_action( 'admin_color_scheme_picker', 'admin_color_scheme_picker' );
    add_action( 'personal_options', 'prefix_hide_personal_options' );
}

function prefix_hide_personal_options() {
  ?>
    <script type="text/javascript">
        jQuery( document ).ready(function( $ ){
            $( '#your-profile .form-table:first, #your-profile h3:first' ).remove();
        } );
    </script>
  <?php
}

Also, a note about the Show Admin Bar option: in 3.3 the admin bar is becoming the header and thus cannot be turned on or off, so it may not be worth worrying about.

I updated the code example to be more accurate and complete. Had some leftovers in there.

As there seems to be no good php hook for this I ended up hiding the fields with CSS then remove them with JS.

add_action( 'admin_head', 'remove_default_profile_fields' );

function remove_default_profile_fields() {

    global $pagenow;

    if( 'profile.php' != $pagenow) return;

    remove_action( 'admin_color_scheme_picker', 'admin_color_scheme_picker' );

    // <tr> selectors, each containing a field
    $tr = array(
        "tr.user-rich-editing-wrap",
        "tr.user-comment-shortcuts-wrap",
        "tr.user-first-name-wrap",
        "tr.user-last-name-wrap",
        "tr.user-admin-bar-front-wrap",
        "tr.user-profile-picture",
        "tr.user-user-login-wrap",
        "tr.user-display-name-wrap",
        "h2" // Personally I decided to remove all H2 tags too.
    );

    $selectors = implode(", ", $tr);

    // Hide the fields with css, so even if javascript is disabled they wont show up. 
    echo "<style>{$selectors}{display:none;}</style>"; ?>

    <script type="text/javascript">
        jQuery( document ).ready(function( $ ){
           // Remove selected <tr>'s
           $( '<?= $selectors; ?>' ).remove();
           // Remove any empty table that may have been left over
           $(".form-table:not(:has(tr))").remove();
        });
    </script>

  <?php
}