In the API documentation, heading "Posts", sub-heading "Updates":
The bullet points look like they were copied from the sub-heading "Create". Especially this can't be right:
post[source] If this is a URL, Ouroboros will download the file.

Anyway, I've tried it, and it doesn't seem to work.
How do I use the API to update the source in
post #155267
to
http://www.furaffinity.net/view/694965/
?

The way I do it is by creating and sending a POST request to http://e621.net/post/update.xml with the following POST fields:

id=<id>
post[source]=<source goes here>
login=<username>
password_hash=<e621.net password hash>

But yeah, post[source] is kind of weird. When uploading, it's used to point to the file you want to upload. (When you're uploading a file through a multipart form, I don't know if post[source] will override it, or set the Source text).

When updating a post, it just sets the Source field like you'd expect. Annoyingly, when you're uploading, since you're already using post[source] to point to the remote file, you can't also set the Source field without sending a second request after the post is uploaded to update the Source field. I fixed this in the next update by adding a post[sourceurl] parameter so you can upload a remote image and set its source in the same request.

I've fixed this in the next update, so you can use post[sourceurl] to fill the Source field if you're already using post[source] to point to the remote image to upload.