Can I have a UIBarButtonItem with a colored image?

There’s other iOS7+ solution:

NSString *iconFilename = // ...
UIImage *image = 
    [[UIImage imageNamed:iconFilename] 
        imageWithRenderingMode:UIImageRenderingModeAlwaysOriginal];
UIBarButtonItem *barButtonItem = 
    [[UIBarButtonItem alloc] initWithImage:image
                                     style:UIBarButtonItemStylePlain 
                                    target:self 
                                    action:@selector(onBarButtonItemTapped:)];

Swift 5:

let iconFilename: String = // ...
let image = UIImage(named: iconFilename)?.withRenderingMode(.alwaysOriginal)
let barButtonItem = UIBarButtonItem(image: image, 
                                    style: .plain, 
                                    target: self, 
                                    action: #selector(onBarButtonItemTapped(_:)))

Extract from UIImage.h:

… navigation bars, tab bars, toolbars, and segmented controls automatically treat their foreground images as templates … You can use UIImageRenderingModeAlwaysTemplate to force your image to always be rendered as a template or UIImageRenderingModeAlwaysOriginal to force your image to always be rendered as an original.