The conjunction "or" is exclusive, which means that if it connects elements (called alternatives) that are both in singular, the phrase is considered to have singular number, which means you need to use "is" in a sentence like
The subject or the object of the sentence is directly dependent...
If one of the alternatives is in plural, it depends on the relative order of the alternatives in the sentence. Choose the number of the closest to the verb alternative:
The teacher or the pupils are liable for the noise in the classroom.
The pupils or the teacher is liable for that racket we keep hearing.
Same goes with other constructs that can stand for both numbers, like "which" or "who":
Is it Paul or his kids who like hiking?
Was it Paul's kids or Paul himself who is coming to live with us?
Other examples and more on subject-verb agreement can be seen, for instance here.
The same agreement needs to exist between the alternatives and any pronoun in a dependent clause. For instance,
Has your dog or cat had its vaccinations?
Your sentence therefore needs to be
For example, the subject or the object of the sentence is directly dependent on the verb and in turn it may take other dependents.