Losing or gaining health (upgrades / downgrades) isn't seen as damage, for example:
- Health up / down pill
- Devil deal
- Black market deal
- Guppy's paw (converts red hearts into soul hearts)
Stuff that deals damage:
- Bad trip pil
- Items that reduces / damages health (IV bag, blood rights)
- Blood banks
- Enemy contact / attacks / projectiles (not all enemies have contact damage)
- (bomb) explosions (unless when you have Pyromaniac)
- Spike door to the curse room
- Devil beggar
Items that help you in a run:
- Holy mantle (automatically blocks the first damage taken)
- Infamy (Can block projectiles on the side Isaac is facing)
- Lost Contact (Block enemy projectiles with your own projectiles)
- Gnawed Leaf (enemies cannot damage you when you're in statue form)
Source: experienced it myself: clearing parts of the game without receiving damage
Beat basement without taking damage.
Beat caves without taking damage.
Beat depths without taking damage.
Beat womb without taking damage.
Beat Chest or Dark Room without taking damage.
- Piercing tears (tears are not destroyed on blocking enemy projectiles)
- Brimstone / Technology (laser will destroy all tears in it's path)
- The Ludovico Technique (floating tear cannot be destroyed, but will destroy projectiles)
- Mom's knife (Knife cannot be destroyed, but will destroy projectiles)
- Midas touch (when enemies hit you, they are turned into gold statues)
- E coli (when enemies hit you, they are turned into poop)
- The virus (when enemies hit you, they are poisoned)
- Gemini (Gemini will auto attack near enemies)
- Ball of bandages / cube of meat (damages enemies near / around you)
- Sacrificial Dagger (damages enemies near / around you)
- Daddy Longlegs (foot stomps enemies automatically)
- Leech (The leech will automatically attack and chase enemy around the room)
- Lil' Haunt (Lil' Haunt will automatically attack and chase enemy around the room)
- Demon Baby (Demon Baby will automatically attack enemies in range)