The story is that Death comes to town and an old grandfather runs
Death up a tree.  While Death is up the tree no one dies and only the
grandfather can let Death down.  The young grandson of the grandfather
is badly injured in an accident and is suffering.  But,
because Death is still up the tree, the boy can't die.  I would like
to know the name of the movie and the main characters.  For some
reason I thought it was one of the Barrymores playing a main role, but
I could be wrong.

Dear magnolia-ga,

I am so glad you asked about this!  This 1939 movie is my most
favorite! It is called "On Borrowed Time" and indeed stars Lionel
Barrymore (in a wheelchair) as the grandfather who traps Mr. Brink
(played by no less than, Sir Cedric Hardwicke)up a tree.  According to
my Leonard Maltin's Movie and Video Guide, the grandmother is Beulah
Bondi and the young girl who falls in love with the Doctor is a very
pretty Una Merkel.  Here is the Internet Movie Database link...

http://us.imdb.com/Title?0031754

and if you need a "heap of tape," (as Pud would say) it is available
on VHS at amazon.com.

I'll see you where the woodbine twine and bring your Kleenex.  Enjoy!
Tan-ga

This sounds an awful lot like "On Borrowed Time," starring--yes!--Lionel Barrymore.

More detail here: http://us.imdb.com/Details?0031754

Hope this helps.