I replaced the chainring on my single speed with one with less teeth. I tried to adjust the wheel in order to get the correct tension on the chain but while rotating the crank the chain would tighten and then loosen depending on the position of the crank. The wheel is centered and the crank seems to be on the tapered shaft of the BB correctly. Could the new chainring be out of round? Has anyone had this problem?

I don't think it is all that unusual. Just find the tightest spot when you turn the crank and adjust the chain for a bit of slack at that spot and you should be good to go. I would also suggest that you invest in a good set of chain tugs if you don't already have a set. They make adjusting the rear wheel position and chain tension very easy. Al

Unless you have a very round, read high end chainring, it's going to do this.

It's not an uncommon issue, usually the problem is that the chainring isn't centred on the crank, rather than that it's out of round. Sheldon Brown explains the usual method to fix it here.

Thanks guys.I understand what is happening. It all makes sense now.

The correct tension for the chain is NO tension. You want the tightest position that always has some slack as the crank rotates. This is a two step process because chainrings are usually slightly (or more than slightly) eccentric, and chain tension varies with crank rotation.

Pull back the wheel until most of the slack is out. Rotate the cranks holding tension by pushing the middle of the lower loop up. (keep fingers to the outside of the loop, ie under the lower loop) and feel for the position of least slack. Keeping the crank there, pull the wheel back until there's only a vestigial amount of slack, enough that you can freely push the center of either loop up or down 1/4". Finally check by rotating cranks and watching and listening to confirm that the chain is never actually tight.

This is sometimes an issue for tandem synchro rings. In that case there are two rings to fiddle with.
Our recumbent tandem uses an adjustable Terracycle idler system for the synchro chain in order to make a turn and adjust tension. Do any fixed gear bikes use idlers to adjust tension? Or would that be adding a usually-unnecessary layer of complexity?

They can't, they put too much tension on the lower run of the chain when the rider resists the rotation of the pedals. If you used one, you'd either break the idler, or the slack would all end up on the top run of the chain whenever the rider was resisting, depending if it was an idler on a sprung arm like a derailleur, or just an idler you adjusted to give the right tension and then locked into place.