Do we know anything about what happened to Sam Witwicky (Shia LaBeouf character) after Transformers 3 or did they just cross Shia off the list and he is never to be seen or spoken about ever again?
Answer for it is given indirectly in 5th film of the franchise, Transformers: The Last Knight.
At one point in The Last Knight, #AnthonyHopkins's character, Sir Edmund Burton, reveals to Vivian Wembley (Laura Haddock) that she's a descendant from Merlin's (yes, the actual wizard) Witwiccan bloodline, which would later be known as Witwicky.
Burton walks Vivian through her extensive family tree, showing pictures of Witwickys throughout the decades. And it's through these pictures that we learn what became of Sam.
The last picture shown from the family tree was of #SamWitwicky — and here's the interesting thing: Burton told Viviane that she's the last descendant of the Witwiccan clan. I'm merely speculating here, but putting two and two together, if Vivian is the last Witwicky that would mean that Sam is ... well, dead.
So his character got killed off screen most probably.