While the Iliad is our earliest source for descriptions of theopanies in the Classical tradition (and they occur throughout Greek mythology), probably the earliest description of a theophany is in the Epic of Gilgamesh. There, the protagonist meets Siduri, a goddess associated with brewing and fermentation. In the Judeo-Christian tradition, the Bible is the primary source of events which both Britannica and the New Catholic Encyclopedia cite as being theophanies.
The appearance of Zeus to Semele in his full godhead, "all his glory", is more than a mortal can stand and she is burned to death by the flames of his power. However, most Greek theophanies were less deadly. Unusual for Greek mythology is the story of the immortal Prometheus, not an Olympian but a Titan, who brought knowledge of fire to humanity. There are no descriptions of the humans involved in this theophany, but Prometheus was severely punished by Zeus.
The New Catholic Encyclopedia cites examples such as Gen 3:8a. The same source then quotes Gen 16:7-14. In this case, initially it is an angel which appears to Hagar, however it then says that God spoke directly to her, and that she saw God and lived (Gen 16:13). The next example the New Catholic Encyclopedia cites is Gen 22:11-15, which states explicitly that it was the angel of the Lord speaking to Abraham (Gen 22:11a). However, the angel addressing Abraham speaks the words of God in the first person (Gen 22:12b). In both of the last two examples, although it is an angel present, the voice is of God spoken through the angel, and so this is a manifestation of God Himself.