AJAX request status 200 but no actual “response”

Your are on the right track but have to tweak a few things.
Your javascript needs (in best case) an object as result and if you want to see the result comming from PHP your js has to handle that. A quick test would be to make a console log – just to check if it works.

Try editing your code like this and check in your browser console:

// PHP file:
function test() {

$result = array(0 => 'Testresult from Ajax 1', 1 => 'Testresult from Ajax 2');
header('Content-Type: application/json');
die (json_encode($result));
}

Then in your js:

// JS file:
function yes() {

var obj = {
    url: ajaxurl,
    type: 'POST',
    data: {
        action: 'test_function'
    },
    dataType: 'JSON',
    cache: false
};

jQuery.ajax(obj)
    .done(function( data ) {
        console.log('result is: ', data);
    });
};

If it works you can handle you results in the done() function, for example write stuff into the dom or whatever.