Wow. This is a complex question. If by "pantheon," you're referring specifically to the Twelve Olympians, then the answer is not much. Much of what we consider "Roman Religion" was heavily shaped by Greek influence on the Etruscans during what are called the Orientalizing and Archaic Periods (around 800 - 500 BCE). Some of the Roman names of Greek gods derive from the Etruscan language (For example, Etruscan Uni and Maris become Roman Juno and Mars, respectively).

If you're referring to how the Roman Empire affected Greek religion as a whole, then the answer is: in way too many ways to describe here.

In short: Partially due to the increase in contact during the Hellenistic Era (after Alexander the Great), religions of subject states were heavily influenced by Greek culture. These influences ranged from specific beliefs or practices, to identities of gods, to religious art. So, on the one hand, you have "eastern" cultures like the Syrians and Egyptians becoming heavily Greek-ified, but by the time the Romans are in power, you have these Greek-ified religions being re-imported into the West.

Romans were extremely interested in foreign religions. During the early Empire, we see the appearance of a cult in Rome dedicated to the Egyptian goddess Isis, which only bears a few similarities to what we know about the actual Egyptian Isis. Isis is just one of the many "Mystery Cults" of the Roman Empire. "Mystery" because only those initiated into the cult know its secrets. The concept itself was native to the Greeks, but during the Roman Empire, there is a proliferation of mystery cults centered around several "foreign" deities.

There was also the Imperial cult, which was dedicated to emperor and his family. Subjects were expected to worship and sacrifice for the health of the current emperor, and in praise of the former (and deified) emperors. This was kind of a problem for the Jews.

There was also the effect of Greco-Roman religion on eastern cultures, such as the Syrians in Palmyra. You've probably heard of the recent destruction of various temples in Palmyra by ISIS. Aside from the tragedy of the loss of any archaeological site, the loss of Palmyra is striking because the religion there was incredibly unique. It represented an amalgam of traditional Semitic religions (gods like Ba'al and Shamash) worshiped in Roman-style temples.

And then there were various folk religions and magical beliefs/practices that are less well-documented, but also reflected this blending of cultures.

I could go on, but like I said, it would take forever. Some good books to check out might be:

Religions of Rome by Mary Beard, John North, and Simon Price

Foreign Cults in Rome: Creating a Roman Empire by Eric Orlin

From Jupiter to Christ: On the History of Religion in the Roman Imperial Period by Jörg Rüpke

Emperor Worship and Roman Religion by Ittai Gradel