Can I apply a CSS transition to the overflow property?

There are many properties that can't be transitioned. overflow is among them; the render engine has no idea how to transition between "hidden" and "shown", because those are binary options, not intervals. This is the same reason why you can't transition between display: none; and display: block; (for example): there are no in-between phases to use as transitions.

You can see a list of properties you can animate here on Mozilla Developer Network.

You can simulate a delay with animation:

$("div").click(function() {
  $("body").addClass("no_overflow");
});

div {
  background: lime;
  height: 2000px;
}

.no_overflow {
  overflow: hidden;
  /* persist overflow value from animation */
  animation: 7s delay-overflow;
}

body {
  overflow: auto;
}

@keyframes delay-overflow {
  from { overflow: auto; }
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div>I'm div</div>

You'll have to apply a separate animation to .body if you want a delay on removeClass, and also to take care that the two animations don't overlap or they'll cancel each other out.

overflow isn't CSS animatable property. You can see full list of animatable CSS properties there.

In case someone is looking at the answer, like I was, for a way to animate the cropping of an element which requires overflowing - here is the solution that worked for me: the clip-path css property which is animatable and very versatile.

Here is a cool tool to play around with, in order to get the proper start / end values for an animation: https://bennettfeely.com/clippy/.