Can you activate a planeswalker's minus ability the turn it enters the battlefield?

Planeswalkers do not have summoning sickness in the way creatures do. The rule against activating loyalty abilities only cares about two things: you must not have already activated a loyalty ability of that planeswalker this turn, and the planeswalker must have enough loyalty counters to pay the cost (CR 606.3).

A minus loyalty ability costs loyalty counters, so the planeswalker must enter with more counters than the ability costs. For example, if a planeswalker enters with 3 loyalty counters and has a −2 ability, you can activate that −2 ability immediately, leaving it at 1 loyalty. If it only entered with 2 counters and the minus ability is −3, you cannot activate it because you cannot pay the cost (loyalty cannot go below 0 from an ability cost).

The once-per-turn restriction means you may activate exactly one loyalty ability of a given planeswalker per turn, regardless of whether it is a plus, zero, or minus ability (CR 606.3). There is no waiting period like summoning sickness.

Concrete example: You play Liliana of the Veil (enters with 3 loyalty) on your turn. You can immediately activate her −2 ability to make each player sacrifice a creature, leaving her at 1 loyalty. You could not then also activate her +1 that same turn.

So the only conditions that could prevent you from using the minus ability the turn it enters are: (1) you already activated one of her loyalty abilities that turn, or (2) she does not have enough counters to pay the minus cost.