What part of Italy still remains independent of the Italian government?

In Italian territory, there are 2 "mini-nations" that govern their territories independently. 

1. Vatican City (Holy See)

Located in the heart of Rome, Vatican City is the home of the Catholic Church, the world's most popular Christian Church. In Vatican City is the Vatican, which is famously home to the Pope.

Relations between Italy and Vatican City have been relatively stable since the signing of the "Lateran Treaty" in 1929 under Italy's then Fascist government.

2. San Marino

Near the Eastern Coast of Italy, San Marino is known as the oldest republic in the world. Within the borders of Italy, San Marino has maintained bilateral diplomatic relations since Italian unification in the late 19th century/early 20th century. 

San Marino is the only surviving municipal state from pre-unification Italy, a territory made up of multiple autonomous city-states.