Get Posts shortcode plugin and meta_query?

Besides the plugin not being updated, this will not work because the meta_query arg is evaluated as a string:

array(3) {
  ["post_type"] => string(7) "project"
  ["meta_query"] => string(96) "array(array('key' => 'state', 'value' => 
'Completed'),array('key' => 'year','value' => '2006'))"
  ["suppress_filters"] => string(5) "false"
}

I suggest you make a custom page template, where you would directly call get_posts() with the meta_query you want.

You will have to copy the code that handles the actual listing of the posts.