Would Loyalty Abilities be negated by Pithing Needle?
Pithing Needle would always nullify the planeswalker's loyalty abilities.

According to rule 306.5d of the November 11, 2016 edition of the Magic: The Gathering Comprehensive Rules, loyalty abilities are activated abilities that (typically) do not produce mana.


  306.5d Each planeswalker has a number of loyalty abilities, which are activated abilities with
  loyalty symbols in their costs. Loyalty abilities follow special rules: A player may activate a
  loyalty ability of a permanent he or she controls any time he or she has priority and the stack is
  empty during a main phase of his or her turn, but only if none of that permanent’s loyalty
  abilities have been activated that turn. See rule 606, “Loyalty Abilities.”


And this is echoed in rule 606.2 of the same compendium.


  606.2. An activated ability with a loyalty symbol in its cost is a loyalty ability. Normally, only
  planeswalkers have loyalty abilities


Additionally, as Jefromi pointed out, loyalty abilities are never mana abilities according to rule 605.1a


  605.1a An activated ability is a mana ability if it meets all of the following criteria: it doesn’t have a
  target, it could put mana into a player’s mana pool when it resolves, and it’s not a loyalty
  ability. (See rule 606, “Loyalty Abilities.”)