Does Possessive pronouns need an apostrophe?

No, possessive pronouns do not need an apostrophe. Apostrophes are used to make nouns possessive, not pronouns.

There are two types of pronouns that show possession:
Possessive pronouns

take the place of a noun that belongs to someone or something.

They are: mine, yours, his, hers, its, ours, theirs.

Examples:

The bluecarishers.
Thehouseon the corner istheirs.
Thechildin the elf costume isours.

Possessive

adjectives

describe a noun as belonging to someone or something. A possessive adjective is placed just before the noun it describes.

They are: my, your, his, her, their, its.

Examples:
Her car is blue.
Their house is on the corner.
Ourchildis the one in the elf costume.

Possessive nouns are formed by adding an apostrophe -s ('s) to the end of the Word, or just an apostrophe to the end of plural nouns that already end with -s (s').

Examples of possessive nouns:

Andy's bicycle

the bank's parking lot

the cat's tail

my daughters' room

the egg's shell

the fence's paint

Germany's flag