We don't know. There are logical arguments to suggest either side.
Firstly, thank you for a wonderful question!
Now, to truly answer this question, it's important to understand who Terence Mann is. Or at least, who he is supposed to represent: J.D. Salinger, the American author most famous for writing the wonderful novel: The Catcher in the Rye.
If you'd like a longer description of this link, take a look at this article. I'll try and summarise it here:
Effectively Field of Dreams is based on a book called Shoeless Joe, by author W.P. Kinsella. In the novel there is no Terence Mann. All the other characters are there, but not Terence Mann. Basically, Kinsella, like many others, read and was fascinated by The Catcher in the Rye and became a huge Salinger fan. In fact, he was such a huge fan that he wanted to include Salinger in one of his novels.
Now, Salinger was very well known for being both reclusive and vehemently against the use of himself or his characters in the media. Despite this, Kinsella went ahead and included him in the book, as a reclusive author who is brought out of his safe place to fulfil his baseball dreams (an admitted desire of Salinger).
Once Salinger discovered his involvement, he was furious. He didn't have enough grounds for a legal case, but made it clear that if the book was ever adapted into other media, he would fiercely oppose it.
To handle this, the film creators made a brand new character, Terence Mann, with the exact same background as Salinger.
So, anything Mann does in the novel, or any of his behaviour, has to be considered knowing all the above information. Kinsella and the film creators knew that his character was effectively J.D. Salinger.
So in light of that, here are some theories:
- Mann is dead most of the film - from the moment he stepped out in front of Ray's car. He had just been dropped off and suddenly appeared to undergo a personality of change of sorts and was desperate to see the field.
This is a fan-theory that's been around for years, that suggests he dies after being dropped off (perhaps a collision?) and that he spends most of the film dead. However, I don't want to go into any more details on it, purely because upon any close inspection it seems to fall apart. After all, his disappearance is reported in the media! Hardly what happens to ghosts. He also calls his father, so, unlike in The Sixth Sense where the protagonist never directly interacts with those around him, Mann is involved and participating in his surroundings.
- Given the fact this whole character is actually Salinger, it was Kinsella (and the film creators) demonstrating the idea of a man stepping out of his reclusive shadow and embracing the endless possibilities the future holds.
This actually makes a lot of sense. At the end, it's unclear exactly what has happened to Mann. He talks like he may come back, but even if he doesn't, who says he can't write? Effectively, he's going to a new beginning, a new start. This is exactly what Kinsella intended when he wrote the character of Salinger (to give him a "second chance"), so it seems perfectly logical that this was his intention for the movie-only character of Mann.
Ultimately, it doesn't matter where he goes, only that he is progressing and moving out of his reclusive shell. The experience of going into the corn is him embracing whatever lies in front of him. He is in someway "embracing death", because he's embracing all the possibilities of life, whatever they may be (including the possibility of death) - as opposed to locking himself away from the world as he was originally doing.
- He walks into the corn knowing he is going to his death, but fully embracing this as he is ready to move on to a different existence
This is very similar to the above theory, but effectively goes one step further. It's not just that he is curious about whatever lies in front of him, but that he knows he is going to his death and actively embraces this fact because he wants to experience something wonderful, different and new. This option is also quite logical. After all, all the other players who come from the corn are dead. And remember this part of the film:
RAY: Wait a second. Why him?
Shoeless Joe and the other players wait for Mann to join
them, ignoring Ray's question.
RAY: I built this field! You wouldn't be here if it weren't for me.
MANN: Ray, for God's sake, I'm unattached. You've got a family.
Mann's last line certainly suggests there is something more sinister at play here - that he is going somewhere beyond human comprehension and it is unknown if he will return. And theoretically, if he did go, he could write his experiences anytime he returned to the field, even if he could never leave it.
We don't know what happened to Mann. The idea he's dead the whole movie seems untenable, but whether he dies at the end or not is certainly up for discussion. There are logical arguments either way, but given the rest of the players on the team, it seems most logical (to me at least) that he did "die" - or at least "pass on" in the way the baseball players did. However, it was left ambiguous for a reason - to allow discussions like this!
You should also note that the book ends the same way. Salinger gets chosen (based on the same interview Mann discusses in the movie, about his dream of playing baseball) and walks through the field, whilst Ray and his wife go the other direction.
Anyway, on a final note, although my head says he is most likely dead, my heart likes to believe it was just a symbolic representation of what Kinsella wanted Salinger to do - to embrace the new possibilities in life and leave his reclusive nature behind him.