From UITabBarItem class docs:
By default, the actual unselected and selected images are
automatically created from the alpha values in the source images. To
prevent system coloring, provide images with
UIImageRenderingModeAlwaysOriginal.
The clue is not whether you use UIImageRenderingModeAlwaysOriginal, the important thing is when to use it.
To prevent the grey color for unselected items, you will just need to prevent the system colouring for the unselected image. Here is how to do this:
var firstViewController:UIViewController = UIViewController()
// The following statement is what you need
var customTabBarItem:UITabBarItem = UITabBarItem(title: nil, image: UIImage(named: "YOUR_IMAGE_NAME")?.imageWithRenderingMode(UIImageRenderingMode.AlwaysOriginal), selectedImage: UIImage(named: "YOUR_IMAGE_NAME"))
firstViewController.tabBarItem = customTabBarItem
As you can see, I asked iOS to apply the original color (white, yellow, red, whatever) of the image ONLY for the UNSELECTED state, and leave the image as it is for the SELECTED state.
Also, you may need to add a tint color for the tab bar in order to apply a different color for the SELECTED state (instead of the default iOS blue color). As per your screenshot above, you are applying white color for the selected state:
self.tabBar.tintColor = UIColor.whiteColor()
EDIT:
