"I'm living a lie" is an expression which turns "living" into a sort of pseudo-transitive verb. "I'm living in a lie" doesn't quite mean the same thing. "I'm living a lie" means that you are saying your whole life is a lie. "I'm living in a lie" means that you are living within a lie, but your life is not a lie in whole necessarily. Think of it like a box, where the box is a "lie" The first sentence is saying that your life is that box. The second sentence is saying that your life is contained within that box. Hopefully that's not super confusing and answers your question, haha. xD :)

There's no preposition because the phrase "to live a lie" doesn't require one; there's no relative location. The phrase works in several examples: Living the dream, living the life... Like Certainly pointed out, you can use prepositions, but they can change the meaning.

Living the good life - another expression. Living in a good way, in a good place. We had a very funny TV series called "The Good Life." It was all about a couple who lived in suburbia but decided to keep chickens and pigs, and grow all their own food. They gave up work and lived "The good life!"

this is poetic language that uses a metaphor. the life is a lie. that is what it is (according to the metaphor). like 'the world is a stage'. you can't add 'in' because there relationship between life and lie is one of equality.