Whe know the word 'her' may play different roles on a sentence.
Regarding its pronunciation, I wonder if it is pronounced differently when it refers to a possessive pronoun or to an object pronoun.
For instance in the following sentences:
John gave her the book. Her book was new.
He gave Mary her book.
Is the word her pronounced the same way in the sentences above?
What I have to say here is from the viewpoint of English in the USA.
When "her" is the first word in a sentence, we pronounce the "h", as in your example sentence "Her book was new". However, in "John gave her the book", it's very common for native speakers of USA English to drop the "h" and say "John gave 'er the book". It's not a rule by any means, and it does not happen all the time, but it is a frequent occurrence.
In the sentence "He gave Mary her book", we would normally pronounce the "h" in "her", as the sound that precedes it seems not to make it convenient to drop the "h". Otherwise, in regular, everyday conversation, we often find the "h" is dropped in "her", as well as in "his", "he" and "him".