Please identify (1) the stage actress who complained, to the effect,
that when doing Shakespeare plays only kings get to sit down and (2)
the text of her witty remark. She presumably played the queen
standing, and my guess is that she was active circa 1920's - 1950's.

The actress was Josephine Hull.

The quote:

"Playing Shakespeare is so tiring. You never get a chance to sit down
unless you're a king."

BrainyQuote: Josephine Hull
http://www.brainyquote.com/quotes/authors/j/josephine_hull.html

In case you can't quite place Josephine Hull, today she is best
remembered as "Aunt Abby" in the classic Cary Grant film "Arsenic and
Old Lace," and as James Stewart's sister in "Harvey."  Both were roles
that she had created on Broadway. Here's a brief bio:

The Oscar Site: Josephine Hull
http://theoscarsite.com/whoswho3/hull_j.htm

And here is a list of Josephine Hull's Broadway appearances:

Internet Broadway Database: Josephine Hull
http://www.ibdb.com/person.asp?id=45928

My search strategy was simplified by the fact that I recognized the
quote from your description. This was the search string that helped me
to locate it online:

Google Web Search: "josephine hull" shakespeare tiring
://www.google.com/search?hl=en&ie=UTF-8&q=%22josephine+hull%22+shakespeare+tiring

I hope this is precisely the information that you seek. If it is not,
or if a link does not function, please request clarification; I'll be
glad to offer further assistance before you rate my answer.

Best regards,
pinkfreud

Ms. Hull whom I saw onstage in Solid Gold Cadillac is the last
Broadway actress from whom I would have expected this devastating
quote.
Congratulations!

grandvenise-ga: Devastating quote? How about juvenile and sour grapes?
monroe22

Grandvenise,

Thank you very much for the five stars and the generous tip! I, too,
found it astounding that Josephine Hull was the source of this quote,
but it has been attributed to her for many, many years. In the
mid-1960s, when I was cast as Gertrude in a college production of
"Hamlet," my director introduced me to the quote, and I have treasured
it ever since. Although Gertrude did get to sit down, it wasn't
exactly a restful sort of sitting. ;-)

~Pink