Wordpress - Removing "Website" Field from the contact info

Revisited and updated answer:

We can't use the user_contactmethods filter to remove the website wrapper, because this piece is hardcoded in the user-edit.php file and not part of the filterable user contacts loop, generated by:

wp_get_user_contact_methods( $profileuser )

Hiding it with CSS

The website row element now got it's own .user-url-wrap class:

<tr class="user-url-wrap">
    <th><label for="url"><?php _e('Website') ?></label></th>
    <td>
        <input type="url" name="url" id="url" 
               value="<?php echo esc_attr( $profileuser->user_url ) ?>" 
               class="regular-text code" />
    </td>
</tr>

Previously we had to use jQuery, to target the parent row of the #url field, for removal.

But now we can easily target the website wrapper and hide it with CSS:

function remove_website_row_wpse_94963_css()
{
    echo '<style>tr.user-url-wrap{ display: none; }</style>';
}
add_action( 'admin_head-user-edit.php', 'remove_website_row_wpse_94963_css' );
add_action( 'admin_head-profile.php',   'remove_website_row_wpse_94963_css' );

Hiding other fields

There are similar row classes:

tr.user-{field}-wrap

available for the fields:

admin-color,
comment-shortcuts,
admin-bar-front,
user-login,
role,
super-admin,
first-name, 
last-name, 
nickname, 
display-name, 
email,
description, 
pass1, 
pass2, 
sessions, 
capabilities,
...

including all the fields from the dynamic user contacts methods.

Here we just replace the {field} part with the corresponding field name.

Screenshots

Before removing the website row:

---

After removing the website row:

I resolved the problem with ob_ functions and DOMDocument. It's better than jQuery or CSS for protecting the form.

I use this kind of solution every time when I can't access a part of HTML content through a hook.

function remove_extra_field_profile()
{

    $current_file_url =  preg_replace( "#\?.*#" , "" , basename( $_SERVER['REQUEST_URI'] ) );

    if( $current_file_url == "profile.php" )
    {
        add_action( 'wp_loaded', function(){ ob_start("profile_callback"); } );
        add_action( 'shutdown', function(){ ob_end_flush(); } );
    }
}
add_action( 'init', 'remove_extra_field_profile' );


function profile_callback( $html )
{
    $profile_dom = new DOMDocument;
    $profile_dom->loadHTML( $html );

    $all_lines = $profile_dom->getElementsByTagname( 'tr' );

    $excludes = array(
        'user-rich-editing-wrap',
        'user-admin-color-wrap',
        'user-comment-shortcuts-wrap',
        'show-admin-bar user-admin-bar-front-wrap',
        'user-url-wrap',
        'user-description-wrap'
        );

    $deletes = array();

    foreach ( $all_lines as $line ) 
    {
        $tr_calss = $line->getAttribute("class");

        if( in_array( $tr_calss, $excludes ) )
        {
            $deletes[] = $line;
        }
    }

    $deletes[] = $profile_dom->getElementsByTagname( 'h2' )->item(0);

    foreach ($deletes as $delete) 
    {
        $delete->parentNode->removeChild( $delete );
    }

    return $profile_dom->saveHTML();
}

Expanding on @birgire's and justifying @Patricia Walton's answer, if you only add

add_action('admin_head-user-edit.php','remove_website_row_wpse_94963');

it will only be gone from the page where admin is editing a profile. To also make it disappear when a user edits its own profile add also

add_action('admin_head-profile.php','remove_website_row_wpse_94963');, like this:

function remove_website_row_wpse_94963() {
    if(!current_user_can('manage_options')){
        // hide only for non-admins
        echo "<script>jQuery(document).ready(function(){jQuery('#url').parents('tr').remove();});</script>";
    }
}
add_action('admin_head-user-edit.php','remove_website_row_wpse_94963');
add_action('admin_head-profile.php','remove_website_row_wpse_94963');