Some heroes such as Clinkz or Outworld Devourer can use auto-cast abilities that add damage to your hero's auto attack,
My question is: does this extra damage count as physical or magical damage?
For example, Clinkz's Searing Arrows add 20 bonus damage at level 1. If your hero's auto attack damage is 50, then are you doing 70 physical damage, or 50 physical damage and 20 magic damage?
Just want to know for the purposes of armour against people like this.
It does the damage type stated in the skill, regardless if it is an auto-cast ability or not. In your example, a Bone Fletcher with 50 base damage and Searing Arrows level 1 would do 70 physical damage per hit and an Obsidian Destroyer (Dota 1) with 100 base damage, 1000 mana left and level 4 Arcane Orb would do 100 physical damage and 90 pure damage.
Auto-casting does not change the damage type the ability inflicts. Also, there are some orb effects that have a little bit extra range when manually cast, such as Viper's.
If you want to find out which damage type a skill does, hover your mouse over the skill and hold Alt. There should be stated in green which damage type it inflicts.
Further to TonySniper's answer, if the attack is set to Auto-Cast, it will count as a normal right-click type attack, and thus be subject to drawing the aggro from towers and creeps. See http://dota2.gamepedia.com/Aggro for more details.
If the ability is not in Auto-Cast mode, and you manually cast it, it counts as a spell cast (but the damage type is not changed), this is important because casting spells will not draw creep or tower aggro. Using this mechanic, you could use say Viper or Drow Ranger to chase someone under and past their tower without drawing the aggro, provided you have a creep wave present attacking the tower and taking the tower's attention. You can also use this to chase past the enemy creep wave while the creeps ignore you.
I think this is the mechanic that you might have been confused with in your question.
By "auto-cast", i assume you're referring to orb attacks (WC3 name) - each orb does damage as described on the description. Almost all do bonus magical / pure (and hence don't contribute to base damage calcs like crit). Another interesting thing is how many different "instances" of damage do they count as - for things like TA's refraction and tree armor.
So for armor, armor won't save you from most of these. BKB's will in almost all cases, especially ones like OD.
Is there any specific orb you want to know about?
Depends upon the ability. It has nothing to do with the auto or manual casting.
From your example, clinkz 'searing arrows' adds physical damage. And Outword destroyer's 'arcane orb' is magical damage although pure.
It means it cannot bypass magical immunity but is a unit is not magical immune it deals pure damage. In 'searing arrows' case, magical immunity has no effect. Auto-casting and orb-walking have no effect what so ever in regards of the type of damage dealt.
