I finally managed to solve the problem,
To achieve this I used a sample code using augmented reality available on Apple developer ressources: pARk
The idea was first to convert the sun's spherical coordinates to cartesian coordinates in order to get its position in the sky as a simple vector: {x, y, z}. The formula is available on Wikipedia : Spherical coordinates. As the distance from the sun (radius in spherical coordinates) doesn't really matter here, I used 1.
Using the device gyroscope and CoreMotion framework I was then able to get the iPhone rotation matrix. From the 'pARk' code sample and using the geometry functions I could then compute the camera projection matrix. I multiply the rotation matrix with it and then multiply the resulting matrix with the sun's vector coordinates.
And that gives a simple vector with screen coordinates for the sun. By displaying a UIView with these x and y values I can finally see the sun moving around as I move the screen.
The code for this feature is available on my Github, feel free to use and share!