Q:

How to have a Mapbox callout marker shown by default and never dismissed?

I would like the annotation title of a Mapbox marker callout shown by default and not tappable. Does the API provide such possibilities? 
EDIT:
Actually what i really need here is: how to show a callout marker to be displayed by default and never dismissed.

A:

To select an annotation and show it´s title without tapping it, this should be enough:
mapView.selectAnnotation(annotation, animated: true)

If you want to show all titles that´s not possible to do, only one is possible.
From your comment, see below update:
Inherit from MKMapViewDelegate and set map.delegate = self in your viewDidLoad.
Add a tap gesture to your map to detect taps outside annotations.
let tap = UITapGestureRecognizer(target: self, action: #selector(tappedOnMap))
map.addGestureRecognizer(tap)

Now we have two functions to work with:
func tappedOnMap(_ gesture: UITapGestureRecognizer){
    mapTapped = true
}

func mapView(_ mapView: MKMapView, didDeselect view: MKAnnotationView) {
    if mapTapped {
        map.selectAnnotation(map.annotations[1], animated: true)
    }
}

When you tap on your map thetappedOnMap function will be called. When you tap outside of an annotation the system will automatically call didDeselect that´s why I have declared a mapTapped Bool variable to keep track if the user has tapped on the map or on an annotation. In the didDeselect you can set which default title should be shown in the if-statement.
So basically, you detect taps outside an annotation and in the didDeselect function you check if it´s a tap on an annotation or on the map. If it´s on the map then you show a default or whatever you like annotation title otherwise the annotation the user has clicked on. You can also always return a default annotation, to do so remove the if-statement in the didDeselect function and just set map.selectAnnotation...
Here is  a sample project that I created for you to try with.

