Depends on what you mean by "burn in". Seriously.
If random images show in the in the vertical bar, you won't see any burn in inside that video area. And no burn in in in black bars, of course.
The problem is that the vertical video area will have more wear than the black bars. You won't notice it at all if you always run vertical games that cover the same video area (aka "at the same res"). But if you display a horizontal game (or normal windows etc), you will see that the black bars area look different than the more faded vertical video area. With modern monitors, it will take awhile to be "bad", but it might bug some poeple even after a short while.
So if you ever plan to use the monitor horizontally, don't be surprised of "burn in", except seen in three large blocks, not the letters/numbers/image burn in most people think of as "burn in".
To sum up:
As others said, the letters/numbers/image burn in is not anymore likely with the vertical on horizontal than normal.
As other said, you will have burn in, but in large blocks not letters/numbers/image burn in, with the vertical on horizontal than normal.