Does Conspiracy remove non-creature subtypes from a creature?
You would retain the Gideon subtype.
Gideon is not a creature subtype, but rather a planeswalker subtype:

205.3j Planeswalkers have their own unique set of subtypes; these subtypes are called planeswalker types. The planeswalker types are Ajani, [...] Gideon, [..], and Yanling.


205.3m Creatures and tribals share their lists of subtypes; these subtypes are called creature types. The creature types are [long list that doesn't include &quot;Gideon&quot;]

Setting the creature type would not interact with the planewalker type:

205.3c If a card with multiple card types has one or more subtypes, each subtype is correlated to its appropriate card type.

If Gideon had become a creature through means that didn't make him retain the planeswalker card type, then he would also lose the gideon subtype:

205.1a Some effects set an object’s card type. In such cases, the new card type(s) replaces any existing card types. Counters, effects, and damage marked on the object remain with it, even if they are meaningless to the new card type. Similarly, when an effect sets one or more of an object’s subtypes, the new subtype(s) replaces any existing subtypes from the appropriate set (creature types, land types, artifact types, enchantment types, planeswalker types, or spell types). If an object’s card type is removed, the subtypes correlated with that card type will remain if they are also the subtypes of a card type the object currently has; otherwise, they are also removed for the entire time the object’s card type is removed. Removing an object’s subtype doesn’t affect its card types at all.