List of Damage Amplifiers

Heroes that have damage-amplifying abilities (not including armor reduction):

• Ancient Apparition (Ice Vortex)
• -15/ -20 / -25 / -30 Magic Resistance to targets in 275 area
• Bloodseeker (Blood Rage - self or target) 
• +25 / +30 / +35 / +40 Damage Increase (Dealt and Taken)
• Chen (Penitence)
• +14 / +18 +22 +26 Damage Amplification
• Elder Titan (Natural Order - aura 275 range)
• Base Resist Reduction: 12% / 19% / 26% / 33%
• Medusa (Stone Gaze)
• Physical Damage taken = +50%
• Magical Damage taken = -100%
• Pugna (Decrepify) - Ethereal effect
• Magic Resist Reduction = 50% (enemies); 25% (allies)
• Reduces most enemy heroes magic resistance to -12.5%
• Shadow Demon (Soul Catcher)
• Damage Amp = 20 / 30 / 40 / 50%
• Skywrath Mage (Arcane Seal)
• Magical Damage Amp = 30 / 35 / 40 / 45%
• Undying (Flesh Golem)
• Damage Amp = 5-20% / 10-25% / 15-30% (scales in 750 range)
• Sceptre Amp = 15-30% / 20-35% / 25-40%

Indirect Damage Amplification:

• Warlock (Fatal Bonds - adds 20% additional damage per additional target)

Items with damage amplification:

• Orchid (30% damage increase, extra damage applied at once at end of Orchid duration)
• Ethereal Blade (40% magic damage increase)
• Veil of Discord (25% magic damage amp)
• Mask of Madness (causes YOU to take 30% increased damage!)

Damage Amplification Formula:

Actual damage = magical damage × (1 + magic amplification) × (1 + second source of magic amplification ) × (1 − natural resistance) × (1 − magic resistance of item) × (1 − magic resistance of first ability) × (1 − magic resistance of second ability)

Natural Resistance = 0.25 (except for 0.3 for Meepo and 0.1 for Visage)

Interpreting the Formula - What does it mean?

In cases where only base hero magic resistance applies, a magic amplification or magic resistance reduction increases the damage by a percentage amount equal to the % amp or reduction.  For example, if a 100-damage nuke is used on a hero within L4 Ice Vortex, it will take 100 x (1+0.3) x (1-0.25) = 100 x 1.3 x 0.75 = 97.5 damage, instead of the expected 75 damage with base resistance, which is an increase of 97.5 - 75 = 22.5 damage, which is an increase of 22.5 / 75 = 30% increase.

There's some additional math (and ability comparison) about damage amplification in this reddit post.

Here's the page on damage amplication on dota2pedia, and a similar page on magic resistance.