<!DOCTYPE html>
<html lang="en-us">

<head>
  <meta charset="utf-8">
  <!--<meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1">-->

  <title> Test</title>
  <meta name="description" content="">
  <meta name="author" content="">
  <meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no">

  <link rel="stylesheet" type="text/css" media="screen" href="css/style.css">

</head>

<body class="">

  <div class="box">
    <div class="box-body">
      <table border="1" id="task" class="table table-bordered">
        <thead>
          <tr>
            <th>#</th>
            <th>Name</th>
            <th>Commands</th>
            <th>Created</th>
            <th>Sent items</th>
            <th>check</th>

          </tr>
        </thead>
        <tbody>
          <tr>
            <td></td>
            <td>1</td>
            <td>GET_MAIL</td>

            <td>2019-09-28 00:13:16</td>
            <td>262</td>
            <td class="forImg"></td>
            <td><button class="viewed">viewed</button></td>
          </tr>
          <tr>
            <td></td>
            <td>2</td>
            <td>GET_MAIL</td>

            <td>2019-09-28 00:13:16</td>
            <td>262</td>
            <td class="forImg"></td>
            <td><button class="viewed">viewed</button></td>
          </tr>
          <tr>
            <td></td>
            <td>3</td>
            <td>GET_MAIL</td>

            <td>2019-09-28 00:13:16</td>
            <td>262</td>
            <td></td>
            <td><button class="forImg viewed">viewed</button></td>
          </tr>
        </tbody>
      </table>

    </div>
  </div>

  <script type="text/javascript" src="js/main.js"></script>

</body>

</html>
.forImg{
  background: green;
}
$(document).ready(function() {
  $("body").on("click", ".viewed", function() {
    // это удаляет класс из кнопки
    $(this).removeClass("forImg");
    // это удаляет класс из соседней ячейки
    $(this)
      .closest("td")
      .prev()
      .removeClass("forImg");
  });
});

External CSS

This Pen doesn't use any external CSS resources.

External JavaScript

  1. https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js