Regina, Saskatchewan; Canada
No they don't, but they originally did. That (■Št) was originally just a neuter form of a the-word meaning "that or the". The genitive/possessive forms for the word were ■Šs in masculine and neuter, and ■Šre in the feminine, and ■ara in the plural (for all genders). The forms for the word this in the same cases were ■isses, ■isse, and ■issa.
The word it doesn't have a different "form" either, but we just add a genitive ending s to it making it its. Originally, though it did have a different form that was exactly the same as the genitive/possessive for he, that is, his! The form his was normally used throughout premodern English for "its".