Possibly. While the film tries to keep the characters relatively realistic, the film still requires us to suspend our disbelief. The left side of Dent's face is soaked in an accelerant, which is ignited, completely burning and charring the skin on the left side of his face. His left eyelid is completely burned as well as most of his cheek, exposing his jaw muscles and the inside of his mouth. While there are no vital arteries to be destroyed, he would be in an unimaginable amount of pain. Perhaps his grief over Rachel's death overshadowed his pain (hence the aforementioned suspension of disbelief). Given the pain, grief and trauma Harvey suffers, it's not difficult to imagine him going insane. Harvey's internal battle, at least according to the classic mythos of the character, is the battle between his evil, revenge-driven half and his sane, compassionate half, hence his new name, Two-Face.
While a person could reasonably survive the burning, it is worth noting that it is not possible to leave the muscles and eyeballs exposed without the tissue dying. Immediate surgery and skin would have been required. Harvey also makes facial expressions that are not possible, since his muscles are so badly damaged. It's also not possible for him to speak as clearly as he does, since half his lips and cheek are missing.
The logic behind Dent's survival exists. The basic form as we know it is the adrenaline rush. The ever popular mother-lifts-car-off-baby scenario comes to mind hand in hand with the fight or flight reflex. The explosion, disfigurement, loss of Rachel, and the existing determination to stop Gotham's corruption all combine in Harvey's mind causing a logically feasible, though relatively temporary adrenaline rush to compliment his fight reflex that allows him to follow his basic, but warped, agenda. His survival would have been directly linked to the duration of the adrenaline rush and denial of medical care. Shock, pain, blood loss, tissue decay and infection in any capacity would have most likely left him dead or incapacitated in the absence of any adrenaline rush. (In fact, as the climax of the movie approaches, the left side of Harvey's suit becomes stained as more blood and ichor from his wound spills onto it.)
