I tested this with
TextEdit and it does not remember the size of it's open dialog window. I tested it with
BBEdit and they both remember the size.
After reading something about another, maybe related situation, I tried this:
I resized the open dialog in
TextEdit like it is described here (second screenshot).
[ which is: hold the SHIFT-key and click into the right side of the dialog and start dragging it ]
And when I do that this way,
TextEdit remembers the size, also between relaunches.
So may be this is possible with all applications you want to remember it's open dialog size? It's not a general solution but a workaround. If it does the job it may be a pretty useful one.
I got the idea to try this from here
(second screenshot): http://osxdaily.com/2014/12/08/resize-large-open-save-dialog-windows-mac-os-x/ and I found that site with a google query containing the title of your question.