Drupal - Get path to theme assets in Twig template

you can use {{ base_path ~ directory }} which will fix the absolute problem, no need to do any preprocessing, both of these variables are included by core.

For example

<img src="{{ base_path ~ directory }}/images/logo.png" alt="My Logo" />

PS. the ~ helper in twig is concatenate.

Edit: at least in the page*.html.twig templates base_path variable is included, possibly you will need to do preprocessing for other templates, you can easily check with {{ dump() }} if the variables are present

// File: THEMENAME.theme in your theme's root directory
function THEMENAME_preprocess(&$variables, $hook)
{
    $variables['base_path'] = base_path();
}

<img src="/{{ directory }}/images/logo.png"/> 

worked for me when

<img src="{{ base_path ~ directory }}/images/logo.png" alt="My Logo" />

did not

for content template, i had to use in .module

   function hook_preprocess(&$variables, $hook) {

  $module_handler = Drupal::service('module_handler');
  $path = $module_handler->getModule('myModuleName')->getPath();

  if(isset($variables['region']) && $variables['region'] == 'content'){
    $variables['module_path'] = $path;
    $variables['http_host'] = $_SERVER['HTTP_HOST'];

and

  <img src="{{ module_path }}/images/error404.png" />
  <img src="//{{ http_host }}/{{ module_path }}/images/error403.png" />

There is by default a {{ directory }} variable that you can use that points to your theme directory. The problem is, it's not absolute, just like the one you added. I think that's a bug in core because it should include the base path, but changing that would of course break existing sites that use it.

So you need to add a / in front of it; this would break if Drupal is installed in a subfolder. You can either hardcode that in your template or keep using base_path() as you did in 7.x in a custom variable.