Spawns a familiar that stabs Isaac at the beginning of each floor if Isaac has more than a Half Red Heart. This drains as much red health as possible without killing him, leaving a half red heart if Isaac only had red health, or none at all if Isaac has soul/black/bone hearts.
For the rest of the floor, Isaac gains extra damage and speed for each heart lost. The amount of damage gained this way is exponential, so each heart lost will provide a bigger bonus than the last.
The exact formula is about +(0.15 * HeartsLost ^ 2) damage and +(0.1 * HeartsLost) speed, where HeartsLost is the total hearts of damage taken from Blood Oath (half a heart counts as 0.5).
Notes
Although the animation is only a single stab when the effect is triggered, it's treated as taking damage one time per half heart lost, leading to synergies with items that activate upon taking damage.
The damage from Blood Oath can be avoided with invincibility, but will not provide any damage and speed increase. That's a useful strategy for the Bloody Mary (challenge #37).
Eternal Hearts give the health up before Blood Oath stabs Isaac, so the new heart is also drained.
Rotten Hearts are only counted as half a red heart each.
XI - Strength can be activated at the start of a floor before Blood Oath triggers to have one more heart counted in the bonus.
If Isaac has half a red heart or less, the effect will not trigger, even if Isaac has soul/black hearts.
Blood Oath also drains the health of Keeper / Tainted Keeper, leaving their health at one coin heart.
The damage caused by Blood Oath does not lower the chance for a Devil Room or Angel Room to appear.
While playing as The Forgotten, red health will not be drained if The Soul is being controlled when Blood Oath triggers.
If Blood Oath is removed (such as with Tainted Isaac), any existing damage and speed bonuses will remain until the next floor.
The damage and speed bonuses are removed after defeating The Lamb and refusing a Victory Lap, or upon coming back from obtaining Knife Piece 2.
Familiars that follow behind Isaac have a certain priority in how close their position is to him over other familiars, with Blood Oath having the lowest priority at 6th and thus being farther from Isaac than familiars of different priorities.
The maximum achievable amounts of stats are +20 damage and +1.15 speed if starting the floor with 12 filled normal red hearts (of which only 11.5 are drained), or +21.8 and +1.2 if there is at least one Bone Heart (which allows all hearts to be drained).
Synergies
Adrenaline: Activates the damage up effect of Adrenaline for the resulting empty heart containers at the start of each floor.
BFFS!: Adds 2 full hearts' worth of bonuses if any health is drained at all, e.g. losing 1 heart gives the bonuses of 3 hearts. This allows for up to 14 hearts' worth of bonuses.
14 hearts' bonus gives +29.6 damage and +1.4 Speed.
Damage-triggered items: Activate a number of times equal to the total half hearts drained.
Bozo: Each half-heart lost has a chance of spawning a rainbow poop, allowing for full red heart regeneration.
Fanny Pack: Each half-heart lost has a chance of spawning a pickup, which can include hearts.
Gimpy / Old Bandage / Shard of Glass: Each half-heart lost has a chance of spawning a full red/soul heart, allowing Isaac to heal back some of the damage immediately if he has high luck or a large amount of hearts lost.
Scapular: Gives Isaac a soul heart if he had only red heart containers upon entering a floor.
Magdalene + Birthright: With full Red Health, Magdalene can achieve up to +46.3 damage and +1.75 speed, and up to +49 and +1.8 with at least one heart being a Bone Heart.
Myosotis: Can transport Red Hearts left on the previous floor, allowing early healing from the damage of Blood Oath.
The Jar: Red hearts can be stored for use on the next floor.
Placenta: With enough patience, health can be safely restored each floor by waiting in the starting room.
Stem Cell: The healing occurs before health is drained, allowing for a greater stat bonus.
Whore of Babylon: Activates the Whore of Babylon effect at the start of each floor.
Interactions
Cursed Eye: The stab will trigger the teleport if tears are charging.
Curse of the Tower: Not recommended. Spawns 6 troll bombs for each half heart removed, typically filling the starting room with troll bombs.
Gnawed Leaf / Tooth and Nail: Being invulnerable when Blood Oath triggers will not drain any health or grant bonuses.
Perfection: The forced damage does not destroy the trinket or count against its acquisition.
Blood Oath originates from the Antibirth mod. In Antibirth, it was possible to prevent Red Hearts from being drained while still gaining the stat increases if Isaac's speed was high enough to immediately leave the starting room. Moreover any Rotten Heart Isaac had would cause Blood Oath to remove half of a Soul Heart which would kill him if he did not have one to remove.
This item originated from the Community Remix mod for the original Binding of Isaac game.
The familiar appears much bloodier when a large amount of hearts are drained.
Bugs
Bug!
Using Glowing Hourglass to return to the previous floor twice will cause Blood Oath to activate an extra time, giving buffs corresponding to drained HP.
Bug!
Upon coming back from obtaining Knife Piece 2 or after defeating the Lamb and refusing a victory lap, the stat boost obtained from this item at the beginning of the floor will be gone.
Half Red Heart. This drains as much red health as possible without killing him, leaving a half red heart if Isaac only had red health, or none at all if Isaac has soul/black/bone hearts.
Eternal Hearts give the health up before Blood Oath stabs Isaac, so the new heart is also drained.
Rotten Hearts are only counted as half a red heart each.
XI - Strength can be activated at the start of a floor before Blood Oath triggers to have one more heart counted in the bonus.
Keeper / 
Tainted Keeper, leaving their health at one coin heart.
Devil Room or 
Angel Room to appear.
The Forgotten, red health will not be drained if 
The Soul is being controlled when Blood Oath triggers.
Tainted Isaac), any existing damage and speed bonuses will remain until the next floor.
The Lamb and refusing a Victory Lap, or upon coming back from obtaining 
Knife Piece 2.
Bone Heart (which allows all hearts to be drained).
Adrenaline: Activates the damage up effect of Adrenaline for the resulting empty heart containers at the start of each floor.
BFFS!: Adds 2 full hearts' worth of bonuses if any health is drained at all, e.g. losing 1 heart gives the bonuses of 3 hearts. This allows for up to 14 hearts' worth of bonuses.
Panic Button trinket.
Bozo: Each half-heart lost has a chance of spawning a rainbow poop, allowing for full red heart regeneration.
Fanny Pack: Each half-heart lost has a chance of spawning a pickup, which can include hearts.
Gimpy / 
Old Bandage / 
Shard of Glass: Each half-heart lost has a chance of spawning a full red/soul heart, allowing Isaac to heal back some of the damage immediately if he has high luck or a large amount of hearts lost.
Scapular: Gives Isaac a soul heart if he had only red heart containers upon entering a floor.
Magdalene + 
Birthright: With full Red Health, Magdalene can achieve up to +46.3 damage and +1.75 speed, and up to +49 and +1.8 with at least one heart being a Bone Heart.
Myosotis: Can transport 
Red Hearts left on the previous floor, allowing early healing from the damage of Blood Oath.
The Jar: Red hearts can be stored for use on the next floor.
Placenta: With enough patience, health can be safely restored each floor by waiting in the starting room.
Stem Cell: The healing occurs before health is drained, allowing for a greater stat bonus.
Whore of Babylon: Activates the Whore of Babylon effect at the start of each floor.
Cursed Eye: The stab will trigger the teleport if tears are charging.
Curse of the Tower: Not recommended. Spawns 6 troll bombs for each half heart removed, typically filling the starting room with troll bombs.
Gnawed Leaf / 
Tooth and Nail: Being invulnerable when Blood Oath triggers will not drain any health or grant bonuses.
Perfection: The forced damage does not destroy the trinket or count against its acquisition.
Soul Heart which would kill him if he did not have one to remove.
Bug!
Glowing Hourglass to return to the previous floor twice will cause Blood Oath to activate an extra time, giving buffs corresponding to drained HP.
The Binding of Isaac: Rebirth
