Weak forms of HAVE

An "object that is not a pronoun" is an object of the verb that is not a pronoun. It's usually a noun or a noun phrase:

I've her concert tickets here,
I've the things you wanted in the trunk of my car.

but it could be other parts of speech as well. See Wikipedia.

Why does a pronoun prevent a weak form?

Probably because pronouns are typically not stressed, and English doesn't like putting too many unstressed words in a row. So if your object is stressed:

I've three tickets to the concert,

you can leave have unstressed, and this lets you use the weak form instead of the full pronunciation. But if you use a pronoun, it's generally unstressed, and that automatically puts some stress on have:

I have her under surveillance.

(Note that the rules about stress in English are more complicated than what I've explained above, and that the stresses I've indicated aren't the only way of stressing the words in these sentences.)