Send reset password link to user from custom lost password form

In your plugin you would first set the redirect URL either to nothing: $redirect=""; or the fully qualified URL of the page you want your user to land on after successfully changing their password. For http://example.com/mypage/ you would use: $redirect=site_url( '/mypage/ ' );

Then your form would be:

<form name="lostpasswordform" id="lostpasswordform" action="<?php echo wp_lostpassword_url(); ?>" method="post">
    <p>
        <label>Username or E-mail:<br>
        <input type="text" name="user_login" id="user_login" class="input" value="" size="20" tabindex="10"></label>
    </p>
    <input type="hidden" name="redirect_to" value="<?php echo $redirect ?>">
    <p class="submit"><input type="submit" name="wp-submit" id="wp-submit" class="button-primary" value="Get New Password" tabindex="100"></p>
</form>

Note: This code is untested. Let me know what happens in the comments if something doesn’t work.

Update

As noted in the comments, if you have a filter on login_url wp_lostpassword_url() will point to your custom page. To temporarily restore the default login_url, remove the filter right before the form code:

remove_filter( 'login_url', 'your_filter_function' );

and add it back right after the form code:

add_filter( 'login_url', 'your_filter_function' );