Doris Duke's Shangri La is an interesting house. Her fascination with islamic-style art is prevalent throughout the house, although only some of it is truly islamic. The tour takes you through several rooms, including the turkish room and the living room, although you do not get to see Ms. Duke's bedroom suite (and since the house was designed around her bedroom suite, it would have been nice to see). The grounds are spacious for Honolulu, and well kept and include a very nice view, although you can get similar views without paying $25.
The house is decorated with several pieces from Ms. Duke's collection, which are interesting, but not overwhelmingly stimulating, and although it's interesting, it doesn't seem like you get $25 worth of a tour since you do not get to see about half of the house.
If you go, go on the 8:30 tour. The house does not have air conditioning and they will give you a small fan which I imagine does not do much good during the hotter parts of the day. Also, the 8:30 tour seems to stay a little longer at the house than later tours.