The prefix jI- is required and the meaning of the following word jIH would change if the jI- prefix was missing (I'll explain that further in a moment). The pronoun jIH is completely optional here and, in fact, a little redundant. Because of that the pronoun jIH is usually only used to emphasize that it was "I/me" and not someone else. If someone asked, nong 'Iv ("Who is passionate"), I might choose to emphasize my answer by saying, jInong jIH, a bit like "Me, I'm passionate."
It turns out there is another completely unrelated noun that is spelled the same as the pronoun jIH. It means "viewing screen". If you were to leave off the prefix jI-, then the verb nong would have the null prefix indicating that the subject is in the third person (neither you nor me, but someone/something else). Thus, in the sentence nong jIH, the jIH cannot be "I/me" because then the verb would have to have the jI- prefix. So it must instead be "viewing screen" which matches with the null prefix. Thus nong jIH cannot mean, "I am passionate," and must instead mean, "The viewing screen is passionate."
qatlho', jdmcowan. I will try to never accidentally make myself out to be a viewing screen! ;)
The subject pronoun jIH is optional here.
A bit like in (say) Spanish or Greek, subject pronouns are optional, since the shape of the verb shows the subject, but it's not wrong to include them. It can give a bit more emphasis.
In Klingon, object pronouns are also optional, since the object is also marked on the verb -- but again, it's not wrong to include them.
For a third-person subject or object, including the subject and/or object can also make things clearer if you want to distinguish between "he/she/him/her", "it", and "they/them".
For example, muvoq could mean "he/she trusts me; it [e.g. an animal] trusts me; they trust me". Saying muvoq chaH narrows it down to "they trust me".
Or vIyaj which could be "I understand him/her; I understand it; I understand them" -- saying ghaH vIyaj narrows it down to "I understand him/her".
And you could even say jIH muvoq chaH or ghaH vIyaj jIH even though the jIH is not necessary.
The verb prefixes, such as jI- here, are not optional.