You haven't actually said what class your Adapter is, which might be helpful. But if it's an android.widget.ArrayAdapter then it has two kinds of constructors - using an array of objects or a list of objects.
Now, you will only be allowed to remove objects from the adapter if the underlying list supports removal. Arrays in Java, as you probably know, are of fixed length (you cannot add or delete an element, only change its value). If I remember correctly the ArrayAdapter internally wraps the array you supply in an unmutable list.
Solution: supply a List instead of an Array to the ArrayAdapter constructor and MAKE SURE that list is mutable (you can use an ArrayList for example or some other implementation that supports .remove() ).
If it's another adapter subclass you're using, then it's difficult to discuss its implementation details and why it won't remove elements if we don't know what it is and how it works.