What orb effects can be stacked in Dota and Dota 2

Solution 1:

As of patch 7.07 : all Orb Effect (Unique Attack Modifiers) have been removed. Lifesteal can stack additively with itself. Only Manabreak (Antimage) doesn't stack with Feedback (Diffusal Blades).

Tehcnically, no Orb Effects stack. That's why they're Orb Effects (or UAM : Unique Attack Modifier in DotA 2). However...

From items

Some items applies a debuff which could be (or was) concidered as an Orb Effect but is not :

Eye of Skadi : "On ranged heroes, Eye of Skadi can be combined with a lifesteal Unique Attack Modifier". So only a lifesteal, no Skadi + desolator for example.
Vladimir's Offering : Lifesteal from Vladimir's Offering is not concidered as an Orb Effect however the lifesteal is reduced on ranged heroes.
Lesser Maim (15% chance to get 20% attack speed and movement speed slow on your target) on Sange (or Heaven's Hallberd) used to be a Orb Effect but no longer is one.
Greater Maim (16% chance to get 30% attack speed and movement speed slow on your target) on Sange and Yasha used to be an Orb Effect but no longer is one.
Manabreak on Diffusal Blades used to be an Orb Effect but no longer is one.

The Orb Effect chain lightning from Mjollnir (and Maelstrom) depend on if you proc it or not (25% chance). When you do not proc it and if you have another orb effect in your inventory, it will be applied. Basically you can lifesteal, corrupt (desolator) etc. between each chainlightning.

That also mean that on a ranged hero you can have Skadi + Lifesteal (Mask of death, Mask of madness, Satanic) + Mjollnir/Maelstrom and a Sange and Yasha and you would benefit from all passive debuff.

From skills

Orb Effect can be provided by some skills in the game aswell. Some of those are passive like Manabreak (Antimage). Some are manually casted or autocasted such as Drow Ranger's Frost Arrow, Viper's Poison Attack, Enchantress' Impetus, Outworld Devourer's Arcane Orb etc.

When dealing with Orb Effect from skills, same rules apply for stacking them (Sange and Yasha works, Vlad stacks too but less lifesteal on ranged hero etc.). For casted one : when you are not using a Manually/Auto casted Orb Effect (like those mentionned above), you can apply another Orb Effect from a purchased item (A common example is Drow + Helm of the dominator : You will want to manually slow your target while you lifsteal between each Frost Arrow).

Solution 2:

The answer by WizLiz already gave you a good overview of item based orb effects, but I want to extend on that a little with skill based orb effects.

Some heroes have attack modifier/orb effect skills, like Viper, Drow Ranger, Outworld Destroyer, Huskar and some more that don't come to my mind right now.

These skills conflict with item based orb effects just like other items would, with one notable exception: You can manually prioritize an orb effect! This is also known as Orb Walking.

The way this works is that you have to turn off Autocast on the hero skill by right clicking it or holding the Alt button when pressing the skill key. You can see whether autocast is on or not by a yellowish box around the skill image.

Once autocast is turned off, the skill will obviously not be applied to your attacks, and your item based orb effects become active. But if you now manually press the skill button and then click on an enemy, the skill based effect will be used instead of the item effect.

This also has another nice sideeffect: If you attack an enemy hero by orbwalking, you will not draw aggro from their creeps!

By using this strategy effectively, you can make use of usually conflicting orb effects, which can give you great benefits.

While this isn't orb stacking per se, it comes near it in many situations when debuffs stay applied for a certain amount of time. Consider the following scenario:

Drow Ranger with Desolator: Drows Frost Arrow skill and Desolator are both orb effects that don't stack with each other. But if you turn off autocast on Frost Arrow and manually use it you can alternate between slowing the enemy and reducing the armor. Depending on your attackspeed you can easily keep up both debuffs at the same time!

Solution 3:

Answers are 3.5 years old and many patches made many changes in orb effects{that is old title in dota2 it is Unique attack modifier}. Unique Attack Modifiers UAM function on a system of priority, because UAM do not stack. For each attack only one Unique Attack Modifier will be applied. In last patches many of UAM was changed to Attack Buffs, Desolator for example. Difference is they will stack with UAM. But still you can have only 1 active UAM. Source

UAM:

The following list contains UAM which do not stack with each other and will adhere to the rules detailed above.

Hero abilities:

Anti-Mage : Mana Break; Outworld Devourer : Arcane Orb

Item abilities:

Morbid Mask, Mask of Madness, Satanic all three items provide Lifesteal and Satanic + Unholy Rage

Attack Buffs.

The following list contains simple attack buffs. These can stack with Unique Attack Modifiers.

Hero abilities:

Abaddon : Curse of Avernus

Broodmother : Incapacitating Bite

Clinkz : Searing Arrows

Doom : Inferal Blade

Drow Ranged : Frost Arrows

Enchantress : Impetus

Huskar : Burning spears

Jakiro : Liquid Fire

Kunkka : Tidebringer

Legion Commander : Moment of Courage

Lifestealer : Feast

Meepo : Geostrike

Sand King : Caustic Finale

Silencer : Glaives of Wisdom

Tusk : Walurus PUNCH!

Ursa : Fury Swipes

Venomancer : Poison Sting

Viper : Poison Attack

Weaver : Geminate Attack

Wraith King : Vampiric Aura

Item abilities:

Blight Stone : Blight

Desolator : Corruption

Diffusal Blade : Manabreak

Eye of Skadi : Cold Attack

Heaven's Halberd : Lesser Maim

Maelstorm : Chain Lighting

Mjollnir : Chain Lighting

Orb of Venom : Poison Attack

Sange : Lesser Maim

Sange and Yasha : Greater Maim

Silver Edge : Lesser Maim

Vladimir's Offering : Vladimir's Aura