Fantasy Island House Appears in this Miniseries
In episode 6 the house Patty Duke is in when here daughter played by Ann Dusenberry falls from the horse is the same house used in Fantasy Island for Mr. Roarke. It's the Queen Anne Cottage at the Los Angeles County Arboretum & Botanic Garden - 301 N. Baldwin Avenue, Arcadia, California, USA.
Currently there are a couple pictures of it here:
http://www.gardenvisit.com/garden/los_angeles_county_arboretum
and more images at Google:
http://images.google.com/images?hl=en&safe=off&um=1&newwindow=1&sa=1&q=los+angeles+county+arboretum+and+botanical+garden+queen+ann+cottage&btnG=Search+Images&aq=f&oq=
Also, in the scenes just before this that take place in Joseph Armagh's office building (the "Armagh Building") where Joseph tells Courtney that he and Ann-Marie Armagh (Ann Dusenberry) are related and cannot marry, these were filmed in the famous Bradbury Building in Los Angeles which has been in many, many films, and which you can see pictures of here:
http://images.google.com/images?hl=en&safe=off&um=1&newwindow=1&sa=1&q=bradbury+building+los+angeles&btnG=Search+Images&aq=f&oq=
It was built in 1893 and you can read more about it (and the other films it appeared in) here:
http://en.wikipedia.org/wiki/Bradbury_Building
More info and pics here too:
http://www.greatbuildings.com/buildings/Bradbury_Building.html
The office building interiors are shown in at least episodes 6 and 7, but possibly others as well.