Neil Patrick Harris started out as an earnest child actor back in the late 80’s. Sometime circa 2004 the world found out that he’s a brilliant comedian, and he never looked back.
Speaking of Leslie Nielsen… 
I wonder how movie history had been altered if he had won the role of Messala in Ben Hur.
Leslie Nielsen BEN-HUR Audition for the Role of Messala!:
John Lithgow who is popularly known for 3rd Rock From the Sun. However he was a well established serious actor long before that, a very talented man.
The reverse seems to be more common. I was surprised to find Ed Wynn – “The Perfect Fool” from vaudebille and many movies (ncluding “Uncle Albert” in Mary Poppins) played several not-a-laugh dramatic roles, including The Diary of Anne Frank and Requiem for a Heavyweight. Lou Costello, of all people, was looking for serious roles, and almost played NYC mayor Fiorello la Guardia. Leo Gorcey, of “THe Bowery Boys”,started out in straight drama (he got his start, after all, in Dead End), and tried to go back.
The same was true of Lloyd Bridges, Robert Stack and Peter Graves in that movie.
Warren Beatty.
But prior to Gannon (who was a comic character), he was in the sitcoms December Bride and Pete and Gladys. Morgan was known for both drama and comedy.
Charlie Sheen
Platoon, then everything else he did.
He probably doesn’t completely fit the profile because Philip Seymour Hoffman has done, and still mostly does, seriously dramatic stuff. But sandwiched in there was a side-splitting performance in an otherwise mediocre, run-of-the-mill romcom, Along Came Polly. I was surprised how funny he was in that and I wish he’d do more comedy.
Schwarzenegger’s first acting roles weren’t exactly what you’d call drama, but they certainly weren’t comedy. It turns out, though, that he’s at least as good and probably better at the comedy stuff as he is at the action.
The back of his head was quite compelling, I’m surprised he did not get the role…
What about the guy who played Thurstan Howell the III (Jim Backus)? I remember hearing that he only had serious roles but he thought Comedy was the highest art form and wanted to do it so bad that he threw himself aggressively at Gilligan’s Island when it came up.
A google search doesn’t seem to support this. While he had serious roles, he also had comedic ones. Oh well 
Beg to differ. Hercules in New York might not be (intentional) comedy, but his role in The Villain surely was comedy. Everything in that movie was the broadest possible comedy.
Yes, on Deadwood he’s… not so funny.
How about Cary Grant/Archibald Leach? Pratfalls and eyerolling double takes in Bringing Up Baby, Arsenic and Old Lace etc. along with being Mr Sophisticated for all time?
What’s more, after Platoon came Wall Street and Young Guns and Eight Men Out – and then came Major League and Hot Shots and Two And A Half Men and so on.
Arnolds biggest pre-*Conan *role was in Bob Rafelson’s Stay Hungry, which was also a comedy - although he himself had a relatively straight role.
(It’s a good movie, too, and not just for Jeff Bridges completists).
[KindergartenCop]It’s naht a toomah![/KindergartenCop]
Well, he was almost exclusively a dramatic actor until “Analyze This” in 1999. Since then, it seems like about half of what he’s done has been comedic (including “Meet the Fockers” and “Stardust”).
I was gobsmacked to see him playing a very camp tranvestite in Stardust.
(Totally excellent movie by the way)
Also as someone else said upthread, John Lithgow, who played a "nasty bit of work "amongst other characters before 3rd Rock.