I would recommend adding a click event listener on the button that checks the active tab and redirects accordingly:

let selectedForm = document.querySelector("#formSelector");

selectedForm.addEventListener('click', function() {
  let activeTab = document.querySelector(".nav-tabs li.active");
  if (activeTab.textContent.trim() == "Google") {
    location.href = 'https://google.com'
  } else {
    location.href = 'https://yahoo.com'
  }
})

<link href="https://maxcdn.bootstrapcdn.com/bootstrap/3.4.1/css/bootstrap.min.css" rel="stylesheet" />
<link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.1.2/css/all.min.css" rel="stylesheet" />
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.4.1/js/bootstrap.min.js"></script>

<div class="container">
  <h2>Dynamic Tabs</h2>
  <p>To make the tabs toggleable, add the data-toggle="tab" attribute to each link. Then add a .tab-pane class with a unique ID for every tab and wrap them inside a div element with class .tab-content.</p>

  <ul class="nav nav-tabs">
    <li class="active">
      <a data-toggle="tab" href="#google">Google</a>
    </li>
    <li>
      <a data-toggle="tab" href="#yahoo">Yahoo!</a>
    </li>
    <li class="pull-right">
      <a id="formSelector" href="#!" target="_blank"><i class="fa fa-plus"></i></a>
    </li>
  </ul>

  <div class="tab-content">
    <div id="google" class="tab-pane fade in active">
      <h3>Google</h3>
      <p>This should change the plus sign icon to google.com</p>
    </div>
    <div id="yahoo" class="tab-pane fade">
      <h3>Yahoo!</h3>
      <p>This should change the plus sign icon to yahoo.com</p>
    </div>
  </div>
</div>

As Spectric said, moving the check to be based on click of a tab would be ideal.

Here's an alternative implementation (requires HTML & JS changes).

HTML

<link href="https://maxcdn.bootstrapcdn.com/bootstrap/3.4.1/css/bootstrap.min.css" rel="stylesheet"/>
<link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.1.2/css/all.min.css" rel="stylesheet"/>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.4.1/js/bootstrap.min.js"></script>

<div class="container">
  <h2>Dynamic Tabs</h2>
  <p>To make the tabs toggleable, add the data-toggle="tab" attribute to each link. Then add a .tab-pane class with a unique ID for every tab and wrap them inside a div element with class .tab-content.</p>

  <ul class="nav nav-tabs">
    <li class="active">
      <a data-toggle="tab" href="#google" related-url="https://google.com">Google</a>
    </li>
    <li>
      <a data-toggle="tab" href="#yahoo" related-url="https://yahoo.com">Yahoo!</a>
    </li>
    <li class="pull-right">
      <a id="formSelector" href="#!" target="_blank"><i class="fa fa-plus"></i></a>        
    </li>
  </ul>

  <div class="tab-content">
    <div id="google" class="tab-pane fade in active">
      <h3>Google</h3>
      <p>This should change the plus sign icon to google.com</p>
    </div>
    <div id="yahoo" class="tab-pane fade">
      <h3>Yahoo!</h3>
      <p>This should change the plus sign icon to yahoo.com</p>
    </div>
  </div>
</div>

Javascript

<script>
    const tabClick = document.querySelectorAll(".nav-tabs");
    const selectedForm = document.querySelector("#formSelector");

    tabClick.forEach((tabEl) => {
      tabEl.addEventListener('click', (e) => {
        console.log(tabEl);
        console.log(e);
        selectedForm.setAttribute('href', e.target.getAttribute('related-url'));
      });
    })
</script>