My Ultraleggera has a Cinelli stem on it, and I cannot get my handlebars to fit - they will not tighten. Is there a shim I need to use, or do I have to use Cinelli bars?

The bike came with some weird Profile bars on it that I tossed - I didn't see any shims in between those bars and the stem.

What do I do?

There are 25.4" bars and stems; there are 26.0 mm bars and stems; and there are 26.4 bars and stems. And now there are even thicker ones. I'm not sure which you have. Aluminum from beer/soda cans makes a pretty good shim, but you have to get it right. It's very annoying to hit the brakes and find your handlebar pivoting forward in the stem.

If is an older Cinelli stem it is most definitely 26.4. I would pick up a Cinelli bar rather than using shims. I have used the beer can method, only to find that hours from home my handlebar has wiggled loose. This happened a couple of times before I gave up on shims, once during an arduous climb.

I just went back and looked at the pix. Did you "toss" those bars as in they are gone forever? If not I would just put them back on - they looked pretty good.

the bars in the photo are the bars I want to use - I assume they are 26, and the stem is certainly bigger.

I have tried the beer can method and couldn't tighten at all - there seems like much more than a .4 mm difference - right now I have several pieces of wire in there, and I get a little bit of rotation - just enough to make the bike uncomfortable when riding the hoods.

The bars I tossed were TT style aero/bullhorns, and they were broken. I also assume that there must have been shims in there that got trashed as well.

The bars in what photo?

Older Cinelli stems are 26.4mm inside diameter as were Cinelli bars outside diameter from the same era.

Most older road bars from the last 20 years have been 26mm diameter but many have also been 1" or 25.4mm diameter, many Nitto bars for instance. You need to accurately measure your bar diameter in the stem clamping area to know what you have and then get a matching stem.

.4mm is .016" which is a lot more thickness than a beer can. You need to get a replacement stem which matches your current bar diameter or get some steel shim material which is .008" thickness and wraps completely around the bar but does not overlap. Probably safer to get a new correct diameter stem presuming the bars are 26mm diameter.

Sheldon Brown has all the answers you seek:
https://www.sheldonbrown.com/cribsheet-handlebars.html