How to structure your WP <nav> tag using wp_nav_menu to use Bootstrap 3's navbar. Note this is specifically for the "fixed top" version of the navbar (http://getbootstrap.com/components/#navbar-fixed-top). Also only works for single-level navs. If there are sub-menus, additional tweaking is needed.

Wordpress Bootstrap 3 nav.php

<nav id="site-navigation" class="navbar navbar-default navbar-fixed-top" role="navigation">
  <div class="container">  
    <div class="navbar-header">
      <button type="button" class="navbar-toggle" data-toggle="collapse" data-target="#main-navbar-collapse">
        <span class="sr-only">Toggle navigation</span>
        <span class="icon-bar"></span>
        <span class="icon-bar"></span>
        <span class="icon-bar"></span>
      </button>
      <a class="navbar-brand" href="#"><?php bloginfo( 'name' ); ?></a>
    </div>
  <!--<a class="skip-link screen-reader-text" href="#content"><?php _e( 'Skip to content', '_s' ); ?></a>-->
  
  <?php wp_nav_menu( array( 
  	'theme_location' 	  => 'primary',
  	'container' 		    => 'div',
  	'container_class' 	=> 'collapse navbar-collapse',
  	'container_id'    	=> 'main-navbar-collapse',
  	'menu_class'      	=> 'nav navbar-nav',
  	'menu_id'         	=> '',
  	'echo'            	=> true,
  	'fallback_cb'     	=> 'wp_page_menu',
  	'before'          	=> '',
  	'after'           	=> '',
  	'link_before'     	=> '',
  	'link_after'      	=> '',
  	'items_wrap'      	=> '<ul id="%1$s" class="%2$s">%3$s</ul>',
  	'depth'           	=> 0,
  	'walker'          	=> ''
    )); ?>
  </div>
</nav><!-- #site-navigation -->