What is the date of composition of the song whose first lines are
"It's only a paper moon Floating up in a Cardboard sky?"

Hello Fie and thank you for your question.

It's only a paper moon floating up in a cardboard sky is part of the
lyrics of the song "It's Only a Paper Moon." whose music was by Harold
Arlen with lyrics by Vincent Rose and Yip Harburg. The full lyrics can
be found here:
http://home.istar.ca/~townsend/early_years/its_only_a_paper_moon_lyrics.htm

According to the official Harold Arlen website this song was composed
in 1932.

“Finally, in 1932 Harold worked again with Yip Harburg and co-lyricist
Billy Rose to write a song for The Great Magoo called If You Believe
In Me. This song was later interpolated into the film version of Take
A Chance, newly titled It's Only A Paper Moon, which in turn became
one of Harold's most well known songs.”

Source: The Official Harold Arlen Website
http://www.haroldarlen.com/bio-4.html

This Yip Harburg biography at the All Music website states that it was
composed in 1933.

“Harburg's main collaborator, however, was composer Harold Arlen. The
duo wrote "It's Only a Paper Moon" in 1933”
http://www.allmusic.com/cg/amg.dll?p=amg&sql=Bcq5ibk096akn

It's Only a Paper Moon from the 1933 Theatre play "The Great Magoo" 
Music by Harold Arlen and lyrics by E.Y. Harburg and Billy Rose, 1933
with Mitchell Ayres' Orchestra
Listen to the music here:
http://home.istar.ca/~townsend/early_years/its_only_a_paper_moon.htm

It's Only a Paper Moon
Lyrics by: E. Yip Harburg
Music by: Harold Arlen
From the Show: Take A Chance, 1932
Listen to the music here:
http://www.owlgang.com/pages/papmoon.htm

The popular song It's Only a Paper Moon has been included on a great
number of albums over the years. Here is the list of albums at CD Now:
http://www.cdnow.com/cgi-bin/mserver/SID=588784561/pagename=/RP/CDN/FIND/popsearch.html/clickID=tn_srch_txt

Additional information that may interest you:

Biography of Harold Arlen
http://www.haroldarlen.com/bio.html
http://www.songwritershalloffame.org/exhibit_bio.asp?exhibitId=53

Yip Harburg’s biography at the Songwriters Hall of Fame
http://www.songwritershalloffame.org/exhibit_bio.asp?exhibitId=14
 
Billy Rose
http://www.centrohd.com/biogra/r1/billy_rose_b.htm

Search Criteria:

It's Only A Paper Moon 
Yip Harburg
Harold Arlen
Billy Rose


I hope you find this helpful and if there is anything that I've
written that needs clarification, please ask before you rate this
answer.

Best Regards,
Bobbie7-ga

What a wonderful answer! Thank you! Fie-ga