Does my base class that inherits from MonoBehaviour needs to be attached to an gameobject within a scene?

It's sometimes pretty hard to understand what people learning inheritance are actually asking because it's so easy to misunderstand inheritance in such complicated ways...

So if you make classes

 class Base : MonoBehaviour {
    public int baseInt;
 }

 class Extend : Base {
    public int extendInt;
 }

what i think you are asking is, do you need to attach both of these scripts to gameObjects to take advantage of inheritance.

The answer is no.

That's because in a sense there are no 'two scripts' here. When you extend a class, it basically means everything declared in the base class gets merged into the extending class when you create an instance of the extending class.

You can almost think of inheritance as a text editing feature instead of anything more complicated.

If you now drag the Extend script into a gameObject you effectively have a script like this on the object

 class Extend : MonoBehaviour {
        public int extendInt;
        public int baseInt;
 }

Related to this, another thing that confuses peolple is that the instances of base script and instances of extending scripts are not magically connected in any way.

If you drag the Base class onto one object and Extend class to another object, then run the game, the scripts are separate instances despite of inheritance. If you change baseInt on one object, that variable doesn't change in the othe objects because they are separate instances. Each of them has their own baseInt despite inheritance.

If you make a base class 'Mammal' and put all the 100 methods and variables required to describe a mammal there and make an extending class 'Horse', that's just a way to avoid having to write those 100 methods and variables again when you make the 'Cow' class. When you make a game with those classes, you probably only attach 10 Cow and 10 Horse scripts to different objects and they will each have their own 100 mammal variables. You don't need to have separate Mammal scripts on objects in the scene to make things work. The mammal stuff is already in each of the cow and horse objects because of inheritance.