One very important point to make regarding your question is that the Phoenicians were not Greeks. They did not speak Greek, they did not worship the Greek pantheon, and they did not engage in activities typically associated with the Greek-speaking world.

The Phoenicians were a civilisation based in what is now the Middle East, in places such as Lebanon, Israel and Syria. They were famous in the Ancient World for being seafarers, and were great traders and colonisers as a result of this. Cities such as Carthage, New Carthage, Sardinian Olbia, Hippo and many others were all originally founded by the Carthaginians. There was not a 'Phoenician Empire' like there was a Roman one, instead, Phoenicia was made up of individual city-states, similar to how things were in Archaic and Classical Greece. Perhaps the Phoenicians greatest invention was their alphabet, based off Egyptian Hieroglyphs, it would be spread across the Mediterranean by their traders, and would ultimately help to create the Greek alphabet, which itself lead to the creation of the Cyrillic and Latin alphabets.

In terms of what happened to the Phoenicians, the Persians conquered the Phoenician heartland in the 6th century BCE, at which point the region was divided up into 4 vassal kingdoms, Tyre, Byblos, Arwad and Sidon. This was not the end of the Phoenicians though, and the region profited greatly from shipbuilding, and became the source of almost all Persian fleets.

Despite a number of rebellions as a reaction to harsh taxes and levies imposed by the Persians, it was not until Alexander I of Macedon conquered the city of Tyre in 332BCE that there was any widespread upheaval in the region. c.2,000 citizens were executed on Alexander's orders as a punishment for resisting his conquest of the city. After Alexander's conquest of the region, Phoenicia was absorbed into first Alexander's Empire, then alternated between various Successor Kingdoms before finally falling under Seleucid control in 197BCE. By this point in time the region was Hellenised, and had also stopped being such an important trading hub.