What would it cost to remake the 1940 song "I've Got No Strings" from
Disney's Pinocchio (originally recorded by Dickie Jones)? We want a
modern band do do a remix.

I cannot answer your question entirely but I hope I can at least help
out some.  Yes the original performer of the song is Dick :Dickie"
Goodman, the voice of the character Pinnochio in the 1940 Disney film.
 The credited song writers are Leigh Harline/Ned Washington.  It is
their estates that you would have to eventually pay, not Disney.  to
the best of my knowledge Disney only owns the rights to the version
recorded for Pinnochio and subsequent use in their company.  If you
give credit to Leigh Harline/Ned Washington you should be fine.  When
your remix gets airplay the familes of those writers will get a cut
for royalties for each time the mix is played.  How much of a cut I am
not sure.  I think that is the way it works.