Velo Dog

Depends a little on where you're riding. If you're on pavement, and it's usually smooth, you could get by with a 28 or 32, though I'd probably use a 35. There's very little, if any, dropoff in speed as you go up a size or two. If your pavement is broken or potholed, or you jump curbs or catch air or ride on dirt and gravel, then 35 or larger for sure, especially at your weight. I weigh 240, and use 35 or 37mm tires for nearly all my road riding. Something like a Panaracer Pasela TG should work fine; I've used them for three or four years and like them a lot. I can run the 37s at 70psi on bad roads without pinch flats.
Whatever you wind up with, don't make the common mistake of overinflating them to "compensate" for their allegedly greater rolling resistance. Part of the reason for larger tires is so you can run lower pressures.
Incidentally, there's usually quite a bit of slack in a rim's tire size recommendations. I've swapped often from 25 to 41mm on the same rims with no trouble at all.