I did the opposite of what you want and put STRAIGHT bars on a Cannondale touring bike...
And YES — it does throw off the geometry when you do something like that and it took me a whole season and THEN riding on a stationary trainer all winter long, and swapping parts around to finally get the position comfortable on the bike. Now I love it.
I call it my "Franken-Commuter-TimeTrial-Touring-ZombieApocalypse" bike.
I took a straight MTB bar, put some old school Onza bar ends on it, and then some clip-on aerobars so it is set up kinda like a TT bike. I have the shimano aero-wheels, an XTR grouppo from an old mountain bike (canti-brakes, drivetrain, etc), but opted for a downtube shift lever / standard brake lever to be able to fine tune the front derailleur on the fly (chain rub) and also to make room for my sweet bell.
But yeah, the job definitely was a bit more than expected out of the box and I ran into lots of issues doing this along the way.
some things to watch out for would be the obvious like stem etc., but ALSO make sure the brake levers are compatible with the brakes you are using. Make sure the brake levers are compatible with the brakes you use and that the levers give enough "pull" — ie., you can't run standard cantilever brakes with a V-Brake lever because there is not enough cable-pull to make the calipers clamp down etc...
Make sure if you use road brake levers work properly with whatever brakes you are using.
There were so many little obstacles, and it was VERY VERY weird riding this bike for a little while (my first ride with it was 400 miles, so I was used to it by the end) — and people who get on it and ride it now are all "WOAH!!!" at first.
I'm totally used to it now and actually have a good riding position but I had to pull the stem out to the maximum height and some other things to make it work. Lots of shims and spending time at Kraynik's to find the right combination of stuff.
Then of course like others said, you need to make sure whatever shifting setup you use will work with the existing drivetrain if you are not swapping it out.
I also swapped the triple crank on this bike with a 42/53 double because I was always having all kinds of bad shifting issues with the tripple and could not find the right front derailleur. I remember using my FOOT whenever I wanted to drop to the granny gear on the crank arm because the derailleur would never bring it down properly before I finally went to a double.
This is actually my favorite bike and the only one I've owned that I would never sell.
Full on touring frame with 3 bottle mounts, front / rear rack mounts / mountain bike drivetrain brakes and levers / shimano aero wheels / and a time trial ghetto bull-horn bar setup.
I am riding this one to Washington D.C. starting tomorrow and I love riding long distances on this bike.
Most people would say "DON'T DO THAT!!!" but it is a TON of fun to experiment with stuff like that in my opinion.