We've only seen Howland Reed once, in one of Bran's visions where he and Ned fought against the Mad King's Kingsguard at the Tower of Joy. This happened approximately 17 years before the beginning of the series.
Why is he never seen in the current timeline of the series?
House Reed is among the most important and loyal bannermen of House Stark and yet never took part in e.g. the War of the Five Kings, although Robb even had to pass through their territory, the Neck.
Howland is the head of House Reed and the Lord of Greywater Watch in the Neck. Currently he remains in his castle.
Before his appearance in the Season 6 flashback he is mentioned only once, in Season 3:
While traveling in the Gift, Bran and Jojen talk about Howland Reed and his relation to Ned Stark. Jojen remembers that when he told his father about his visions of what had happened to Ned Stark, it was the first time he saw his father cry.
His only involvement in the War of the Five Kings was sending his children to aid Bran. Why there was no military action from his side is unknown, although it's not that surprising if you take into account that the crannogmen are known for keeping almost no contact with the outside world and aren't the greatest soldiers or warriors.
There isn't much more regarding Howland in the books, apart from a very small plot:
In the third novel, Robb Stark sends Maege Mormont and Galbart Glover to the Neck, to inform Howland Reed about his plan to retake Moat Cailin from the ironborn; Robb is certain that Howland will not let him down. Robb is killed, however, before the plan can be realized. It is unknown whether Maege and Galbart reach their destination.
When asked if Howland Reed will come out in the future books George R. R. Martin said:
He will appear eventually.
True, we didn't see him, but he was briefly mentioned though
In s03e02, Dark Wings, Dark Words, when Bran and Jojen were talking about Ned stark's death
B: When my father died, I dreamt it.
J: You didn't dream it. You saw it. So did I.
B: You have the sight, too?
J: When I told my father about your father...for the first time in my life, I saw him cry.
B: Your father is Howland Reed?
J: Yeah.
B: He saved my father's life during the rebellion.
J: Your father told you about the rebellion? Mine never did.
So, what we know is that he's still alive (or at least was by the time Ned died), He still consider him a friend and he don't talk about the rebellion.
To expand a little on Chandler's answer. Crannogmen indeed do not go much outside of their swamps (you know, they stick to Cthulhu and stuff ;) ). When you talk about Reeds and Greywater Watch, you don't picture normal castle and feudal lords. Their castle is known to move and no enemy can conquer them because Reeds and crannogmen will move through swamps and marshes and avoid the fight. So it is surely not castle like citadel with walls and Reeds don't have soldiers and shiny armors and swords.
It's been a while since I read the books (only the first three books) so someone may correct me. Howland was actually personally interested in seeing the world outside, so decided to travel around on his own and went to Harrenhall tournament, seeing Starks in person. He was the son and heir to the lord, but like I said, he's crannogman. He looked in a way that servants to some southern lords felt free to harrass him, something they wouldn't even dream to do to someone on higher feudal level. So when the Robert's rebellion came, Howland went to help the Starks in battle and is either the only one of the crannogmen or perhaps there was a handful of his men with him, but surely no armies because that is not the way of his house nor people. After this he stays in his lands and like other of his countrymen and doesn't communicate with outside world. It is unsure what crannogmen actually pay as tribute to Starks, but it is surely not soldiers nor steel, and probably no gold either. Some food, most likely.
So, when reading the book and not suspecting that Brandon may actually get to see and learn everything, it was clear from storytelling point of view that you would keep the only witness of Tower of Joy events besides Ned for sometime later in the story. He surely knows some secrets and it was interesting enough to see his children join up with Brandon, seemingly knowing or suspecting something is off with the whole story. That is something we know from the show as well and we learn from his children that Howland is alive and well.
So, short answer is: nothing happened to him, he's alive and living in his lands, not appearing because of crannogmen habits and because whatever secret he knows he won't easily let others find out.
