Burn
B/W Tokens is generally favored against Burn. This build runs seven mainboard lifegain options, and ample removal for all of their creatures. Unfortunately, Tokens also tends to bleed life quickly and some of our cards like Thoughtseize and Bitterblossom are pretty terrible in this matchup. When Burn is on the play with a good hand, Game 1 can be pretty difficult, but resolving Sorin with an active board state or delaying long enough to activate Vault of the Archangel will almost always seal the deal in our favor. Nevertheless, getting burned out before then is a possibility.
Essentially every B/W Tokens build runs ~2 sideboard Burn hate-cards. I prefer Auriok Champion, in part because of its versatility in other matchups, but Blessed Alliance and Timely Reinforcements are definitely both viable. Postboard games are very difficult for Burn for several reasons. Once we know what we are playing, we will almost always keep a hand containing at least one strong lifegain opportunity, we will be more cautious with how we utilize our land base, we will have ditched low-value cards like Thoughtseize, and we are now armed with game-ending bombs like Timely or Auriok. Collective Brutality and Liliana of the Veil are also both fantastic in this matchup, and we move in a second Collective Brutality postboard to make things even more difficult for them.
In my experience, B/W Tokens is favored about 65/35 against Burn.
Affinity
B/W Tokens has a particularly strong matchup against Affinity. The combination of the large number of flying creatures and the strong removal suite B/W Tokens runs makes it inherently difficult for Affinity to execute a successful gameplan. That's not to say that it's impossible though. Affinity has two primary factors going for it that can lead to game wins in this matchup: nut-hands that are so fast that Tokens doesn't have time to mount a defense, and absolutely devastating sideboard bombs like Ghirapur AEther Grid. More often than not, though, Tokens will take the match, in part because we run strong sideboard cards of our own for this matchup, like Stony Silence, Pithing Needle.
Based on my own records and general consensus about the matchup, B/W Tokens is favored about 75/25 against Affinity
B/R Hollow One
Hollow One is a counterintuitive matchup in which B/W Tokens is distinctly unfavored pre-board, in part because we are running about a dozen cards that are either near-useless, useless, or potentially even worse than useless against their deck. However, we are running so many sideboard cards that are excellent in this matchup that Tokens is distinctly favored postboard. Let's discuss the pre-board game first.
There is a significant degree of variability inherent to Hollow One, and it is possible for them to put so much power on the board in the first few turns that we (or any other fair deck) will simply be unable to handle it. More often than not, however, they won't "god-hand" us outright, but will apply enough pressure that it can be difficult for Tokens to stabilize. Path to Exile is an all-star card in this matchup, and all of our two-drop options are valuable or important in one way or another. Unfortunately, hand disruption is generally terrible here (exception being made for Collective Brutality), and Liliana of the Veil is useless unless their board state consists of nothing but one non-recursive creature, which is almost never the case by Turn 3 in a pre-board game. It is important to play token producers early and frequently if you're going to have a chance at stabilizing and eventually becoming the aggressor, but also important to hold up removal when prudent. Even a single Flameblade Adept can swing in for 7+ damage when we're unprotected. Don't fret if you get your face beaten in Game 1 by Hollow One. That is what the deck is designed to do, and it does it well. You'll have a very good chance of making up for an initial loss in games 2 and 3.
The postboard matchup against Hollow One is an entirely different story because we switch out all of our crap (for the matchup) for powerhouse cards that either dramatically assist in stabilizing, or neuter their deck in one way or another. In truth, a hard Wrath like Damnation or Wrath of God would be great here (and Settle the Wreckage would be even better), but it's OK that we don't have those because our other sideboard cards are so good. Auriok Champion is indestructible, serves as an infinite blocker for Angler and Tasigur, kills Bloodghast, and can double-block with a token to kill Flameblade Adept. Additionally, it provides a ton of lifegain in this matchup, which helps bide us time to stabilize. Collective Brutality provides us with hand disruption that is not going to accidentally force Bloodghast or Phoenix into their graveyard, can kill Flameblade, and can provide us with lifegain if we're against the ropes. Surgical Extraction is basically broken here. As soon as they get Bloodghast or Flamewake Phoenix into their graveyard, we can freecast it to knock out every one they have in their 'yard and hand, while simultaneously reducing their deck's threat density significantly. Nihil Spellbomb can simultaneously exile several of their threats. I have used it to take out four at a time before, and the fact that it also cantrips us into another card is icing on the cake. It also makes it more difficult for them to cast Delve creatures. I also like to throw in Anguished Unmaking as essentially bad Path to Exile number five. Anything that can exile their recursive threats is good here. If some combination of these sideboard cards isn't in your opening seven, it is often worthwhile to try to mulligan into one. It's hard to stress enough how valuable they are.
Based on 30+ games against B/R Hollow One, I would say that they have roughly a 60/40 advantage Game 1, and the odds turn in our favor 70/30 games 2 and 3, putting the overall matchup at about 55/45 in our favor.
Merfolk
Merfolk is one of those matchups that can easily go either way depending on draws and player skill. To be able to deal with a deck that simultaneously goes big and goes wide, having the right tools for the job is essential. Pre-board, having one disruption spell and two removal spells is an ideal place to start. When on the play, ripping Aether Vial out of their hand to slow Merfolk down can go a long way towards establishing a moderate board state without being run over, and having that time to draw into additional removal can be key.
Sometimes they will get Vial out ahead of our ability to do anything about it, so keeping mana open to kill their Lords is important--either normally cast or Vialed in. This all kind of leads to the fact that we are usually going to have to adapt our play style to what they do, essentially play a reactive game unless we have the luxury of taking out all of their lower mana threats with hand disruption and early removal. If we pull their Aether Vial and can afford to take some level of early-mid game damage, it is possible to race in this matchup, although everything has to line up. Disruption into Intangible Virtue into Spectral Procession into Sorin, Solemn Visitor on the play can put BW Tokens in a position in which we start to gain life faster than they can deal it, which brings me to mention one of the most powerful cards in this matchup: Vault of the Archangel.
If Merfolk can be slowed down enough to get Vault online, it puts them in a really bad spot, because we are generally very happy trading tokens for their highest value targets-- and gaining life in the process. Once we have Vault up, it sometimes makes sense just to leave mana open and pass turns until we draw into the ability to either create an army that they can't stop, or (if we haven't already) play Intangible Virtue and start wearing them down while still being able to keep our tokens open as Deathtouch/Lifelink blockers. One thing to watch out for is a topdecked Spreading Seas, which often spells death for us if we have not been shooting down all of their Lords with Push and Path. Early-game, it's still a dangerous card, but unless it totally mana-screws us, we can generally play around it; the key is holding up a removal spell for Master of the Pearl Trident, however tempting it may be to fire off at one of their other creatures.
Postboard, there are some key cards we want to move in, like Anguished Unmaking to deal with Aether Vial and Spreading Seas, and Flaying Tendrils for the early-game one-two punch of EOT removal targeting their Lord followed by Tendrils to wipe the board clean.
Overall, I would say that Merfolk is a slightly unfavorable matchup for BW Tokens, about 45/55 based on my own data and discussion with other players.
Elves
Elves is a solidly bad matchup for BW Tokens, and all other things being equal, we are going to lose significantly more games than we are going to win. The problem essentially boils down to the fact that although both decks in this matchup utilize swarm strategies, Elves is much, MUCH more explosive. While our spot removal is very valuable here, Elves can usually pull more high value creatures through Collected Company, Chord of Calling, and an exceptionally high creature density than we can deal with. Disruption is our second most valuable tool in this matchup. Inquisition of Kozilek hits all of their creatures, and Thoughtseize can get rid of CoCo, a card which has a habit of undoing our early-game efforts.
Postboard, the strongest play we can make against Elves is the combination of spot removal targeting a Lord followed by Flaying Tendrils, which is even more devastating in this matchup than it is against Merfolk. Played well, it essentially guarantees the win. Even Zealous Persecution is very strong against them. Unfortunately, we can't always rely on drawing a sweeper, so Elves still has the edge postboard.
Based on my experience with the matchup, I would say that odds are about 30/70 vs. Elves.
Bogles
The Bogles matchup largely depends on the opening hands kept by each player, as well as which player is on the play. Because Bogles frequently keeps hands with only a single creature, one mana disruption can be incredibly valuable for us, and devastating for them. Spot removal is useless obviously, and some of our cards like Bitterblossom and Collective Brutality are pretty underwhelming. Liliana of the Veil really shines here, and a hand with Inquisition/Thoughtseize + Liliana will essentially guarantee us the win. Postboard, we take out our removal, move in Anguished Unmaking, Slaughter the Strong and few filler cards, and hope that they don't get Leyline of Sanctity.
Although it is build dependent, I would say that Bogles is slightly favored on balance. Our odds are roughly 40/60 for this matchup. (note: we are significantly less favored against the latest version of Bogles that runs 4x mainboard Leyline. It has in fact become a problematic matchup.)