Ich bin diese Debatte leid

The subject of a sentence is always in nominative case. That other parts of speech are also in nominative case is rare. This only can happen if the verb is a copula. Such verbs are:

• sein (to be)

  Michael ist Lehrer.
  Michael is a teacher.

• werden (to become)

  Laura wird Mutter.
  Laura becomes a mother.

• bleiben (to stay)

  Susanne bleibt Autobesitzerin.
  Susanne stays an owner of a car.

Sometimes also some other verbs can work this way. But if so, then the sentence always expresses that something is equal to something else, or was equal or will be equal. The official term is Gleichsetzungsnominativ (nominative of equalization)

If the sentence doesn't express such an equalization, then the object is not in nominative case.

It always depends on the verb in which case the object has to be. Only copulas (to be, to become, to stay plus some rare constructions with other verbs) need their "object"¹ in nominative case.

There are 51 German verbs that need their object in genitive case (bedürfen, gedenken, entledigen, ... full list is here). All other verbs need dative or accusative case.

1: _{Technically an "object" in nominative case, as discussed here, is not really an object. It is a part of the predicate. But still at fist sight it may look like an object.}

As a rule of thumb you can assume objects to be in either dative or accusative. Some expression require a genitive object. A nominative object¹ is used only rarely as in

Ich bin ein Mann

Your example sentences need an accusative object. If this sentence had been written with a male object, this would have been clearer, since the male nominative and accusative form aren't homonyms.

Ich bin diesen Mann sowas von leid!
Ich habe diesen Regen sowas von satt!

¹ as a technicality, it's not really an object but a predicative nominative.