Nowhere on the Psychic section does it state that Psychics are arcane or divine casters.
As such, it's not clear if they suffer spell failure, let alone whether or not they can prestige into Dragon Disciple.
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This does not appear on the spellcasting ability, because it later appears on the Psychic Spells class ability, few pages later (Occult Adventures, page 69):
Psychics gain access to the following spells. Many of these spells are found in the Core Rulebook; the spells marked with an asterisk (*) appear in Chapter 4 of this book, and those that are found elsewhere are marked with superscript abbreviations denoting their source. The psychic casts all spells as psychic spells.
At least, not without taking levels on another class that grants arcane spellcasting. This is because Psychic Magic is different from arcane and divine magic.
Wizards study ancient tomes to unlock arcane secrets of the universe, and clerics pray to distant deities to grant them divine power. Yet there is a third, more esoteric kind of magic, connected to every creature's composite being, from the conscious mind to the deepest desire, from the life force to the spirit, from the very soul to the cosmic self. This third type of magic is psychic magic.
That said, psychic spells do not use Somatic Components, which is necessary for the Arcane Spell Failure mechanic to work:
Instead of verbal and somatic components, all psychic spells have components related to the caster's inner being. The two psychic components are called emotion components and thought components.
The rule for Spell Failure is exclusive to arcane casters. Since psychic spells are not arcane, psychic spellcasters do not suffer any penalties on their spells from wearing armor.
Casting an Arcane Spell in Armor: A character who casts an arcane spell while wearing armor must usually make an arcane spell failure check. The number in the Arcane Spell Failure Chance column on Table: Armor and Shields is the percentage chance that the spell fails and is ruined. If the spell lacks a somatic component, however, it can be cast with no chance of arcane spell failure.
As you put occult adventures into your tags I take it it is the psychic from there (there are a few other types of psychics around already). Now we have to look at the wordings: http://www.d20pfsrd.com/occult-adventures/occult-classes/psychic for the psyhic itself and the dragon disciple class: http://www.d20pfsrd.com/classes/prestige-classes/core-rulebook/dragon-disciple