Answering the original question, however (as I seem to be just into writing posts here tonight) there are a several factors that weigh into a Civ being consider sub-par, including:
1) Being too Map-Dependent (this is usually more of an issue for naval-focused civs, for obvious reasons, though I'd argue that civs like Mongolia, Assyria, Huns, etc. can face this the other way.)
2) UA is generally unhelpful, for any number of reasons.
3) UU comes too late to be useful.
4) UU doesn't carry benefits over on upgrade.
5) UU has downsides that cancel out the upsides too much.
6) UU upgrades into an undesirable unit.
7) UU's benefits are terrain-based.
8) UB replaces a Building you wouldn't normally want.
9) UB has downsides that cancel out the upsides too much.
10) Bad start-bias.
And I'm sure there are others I'm not thinking of. There are also issues of a UA having downsides that cancel out the upsides, but in those cases, usually the issue is of people just not digging the downside/upside, and not seeing the potential powerhouse there for the taking (e.g. India, Ethiopia, Venice) so I'm not really counting those.
But if a Civ has several of these issues, it's going to be a problem.
Denmark for example, is usually considered among the worst. It is very heavily map-dependent, as it basically demands amphibious attacks to make it work. It works pretty great on an ideal map, but otherwise? Meh. The Berserker and Ski Infantry are both terrain-based, in one way or another. The Berserker is arguably still pretty good - +1 movement and comes early - but it all adds up to a civ that does one highly situation thing pretty well, and has no focus or benefits elsewhere.
Byzantium is another one that faces a lot of problems here. One of the UUs - the Cataphract - is slower and stronger than the Horseman at a point in the game where speed is more important than strength in most cases. (See also: African War Elephant.) Potentially there is a Cataphract-swarm strategy out there, but I've never come across it.The other UU - the Dromon - is actually one of the best UUs in the game, but can't do enough for Theodora to make the Byzantines worthwhile as a whole, and obsoletes entirely upon upgrade. And that is because of the well-intentioned, worthless-in-practice UA. Getting an extra religious tenet sounded great when G+K was first coming out, and must have sounded great to the designers, so much that they split the two halves of this that could have worked together into two separate civilizations, with the Celts getting the much better end of it. The big problem with the Byzantine UA is that when you're on a low-enough level to get a religion, the bonus makes a negligible difference. When you're on a high enough difficulty to have the extra tenet really make or break things, you're almost certainly not getting a religion anyway, so the UA does nothing much, or nothing at all. With the addition of Reformation Beliefs, that diluted the Byzantine UA even further, making it one of the worst, if not the single worst, in the game.
**A special note here about Mongolia, whose UA battles Byzantium's for Worst in the Game, though Mongolia is a great civ in total. Conquering City-States is something you never really ever want to do in the game, yes, but as an AI benefit, the flavor works great, and Keshiks and Khans are so great that it doesn't matter if the player never makes full use of the Mongols' UA. If AI Byzantium founds a religion, though? Who really cares about the extra tenet? That's the difference, to me.**
**Another special note here about flavor - the Byzantine UUs are ancient and classical. The Byzantine Empire's existence basically defines the timeframe of the Medieval period. This bugs the hell out of me.**
Japan is better in BNW following their patch, for sure. I actually wouldn't put them near the bottom myself, but I understand it. For one thing, the Zero is built for a game with air-battles that just don't ever really happen in my experience in Civ V. Removing the oil requirement makes it better, for sure, though there's a flavor issue there as well (in that the Zeroes caused major issues for Japan in WW2 over oil resources.) The Samurai is a good unit, not great, made synergistic and nice for Japan, in peacetime. Building fishing boats, which themselves generate culture for Japan, allows for versatility. Nothing to write home about, but good. But then there's Bushido. The Japanese UA looks great to newcomers who expect the attack penalty for wounded units to be far greater than it actually is. The culture benefits from the UA can be wonderful - Japan is the only Civ that can plausibly expect a culture boost from Turn 0 - but it takes a lot of re-rolls to found near an Atoll in my experience. Still, I think Japan has been moved post-patch from bottom-tier to simply lower-tier. Not great, but far from the worst.
Brazil hasn't really been brought up here, but I'd say that their jungle-start, combined with nothing to mitigate it in the early game (a la the Aztecs, who I think are top-tier or near it) make them deserve a spot in this discussion. Low-production starts are my bane, the Brazilwood Camp doesn't do anything to help, and the Pracinha comes too late to be worthwhile. Considering the hammers necessary for a culture victory, which is all that Brazil is built for, I'd argue that they come into the game with a major handicap.
Finally:
Iroquois have one okay UU, which nobody really thinks about when thinking about UUs, are extremely and unreliably map-dependent, and have a UB which is arguably worse than what it replaces, and hampers tile development in a lot of ways (compare with the Ducal Stable, which encourages it.) Basically, the Iroquois have to be compared with the Inca, who are basically superior in every way, and are Top-Tier in my opinion. Hills last forever. Forests last until they don't. Mohawk Warriors get a terrain bonus. Slingers get one based on UA and also a tactical advantage. Plus the Incan economic benefit lasts the entire game while, again, forests probably don't.
In the end, I think the award for "worst" has to go to either Byzantium or the Iroquois. Because Science and Analysis.