Can you use a 1031 Exchange to purchase a foreign property (which is held using a California TIC structure)?  We are selling a fractional apartment in Paris that is held first by an SCI (french real estate vehicle used to own a property with multiple, unrelated owners) and then by a California TIC.  I've had several inquiries about using a 1031 exchange when purchasing.  First of all, is this even a possibility?  Then what types of exchanges would qualify?

No, 1031 exchanges require that all of the properties involved in the 1031 exchange must be like-kind to each other.  Like-kind means that all of the properties must be domestic (U.S. property) or they must all be non-domestic (foreign properties).  You can not 1031 exchange between domestic and non-domestic properties.