If the photograph is a lie or not, in my opinion, depends clearly on the author's intention and it doesn't have anything to do with the photographed subject, methods and techniques.
Similarly to the written or spoken word or any other means of communication, a photograph can be indeed a a tool for a manipulative practices, we can’t do anything about it.
It is up to us to make an effort of finding out what is the context and author intention if we want to understand it. The other option is to have an personal interpretation of it but then we need to be aware that very often our understanding is far away from the true motives which determined the existence of that particular image.
Constructed reality as an subjective expression is an creative communicate as any other - drawing, poem, melody etc. It is though much more difficult to interpret due to the layer of obvious connection to the real word, hence the context and author statement is very important.
If Crewdson stated that his meticulously constructed scenes are real that would be a lie. Would anyone believe it? Thats a different question…
Sugimoto’s photographs of American Museum of Natural History visualisations (Diorama series) are an interesting case. He did not constructed the scene he photographed, it was constructed though to represent some speculative reality by the museum designers. In the ontological sense, the record is an authentication of that scene existing in real but the scene itself is a visualisation and Sugimoto knows about it. So he intentionally records a fiction but he is clear with the audience about that. It would be hard to believe in the authenticity of the existence of some subjects but most of the photographs create the strong illusion of reality.
I had found a way to see the world as a camera does. However fake the subject, once photographed, it's as good as real. (Hiroshi Sugimoto)