If you read Godless Shrine's ability carefully, you'll notice that it doesn't say the card enters the battlefield untapped if you do pay; it says the card enters the battlefield tapped if you don't pay.
Godless Shrine only affects how the card comes into the battlefield when you don't pay; it doesn't affect how the card comes into the battlefield when you do pay.
Flagstones of Trokair causes Godless Shrine to come into play tapped, and Godless Shrine doesn't do anything about that (even if you do pay).
It would come into play tapped as a ruling on the card says other abilities take precedence. If you look at the Gathering page for Godless Shrine you will find your answer.
If another effect (such as Loxodon Gatekeeper’s ability) tells you to put lands onto the battlefield tapped, it enters the battlefield tapped whether you pay 2 life or not.