Can you respond to a planeswalker's loyalty ability after it is activated?

Actually, the answer is yes — you can respond to a planeswalker's loyalty ability after it has been activated. Loyalty abilities are activated abilities that use the stack just like any other activated ability (CR 606.1). Once the controller activates the ability and it is placed on the stack, both players receive priority and may respond before it resolves (CR 116.2).

The key restriction is not about responding, but about activating: a loyalty ability may only be activated once per turn, and only on your own turn, at sorcery speed (CR 606.3b). Also, the loyalty counter cost is paid immediately as part of activation — removing those counters is not something you can prevent by responding (CR 606.2).

So if an opponent activates Liliana of the Veil's −2 ability, you absolutely may cast a spell or activate an ability in response. For example, you could cast Hero's Downfall to destroy Liliana in response — the planeswalker will be destroyed but the loyalty ability is already on the stack and will still resolve (since its source no longer needs to exist for a triggered or activated ability to resolve — CR 112.7a).

Common ways to interact in response include: countering the loyalty ability with a spell like Disallow, destroying or bouncing the planeswalker (though the ability still resolves), or using your own instant-speed effects to set up before the ability's effect happens.

• CR 606.1: Loyalty abilities are activated abilities.
• CR 116.2: Players receive priority after an ability is placed on the stack.
• CR 606.3b: A loyalty ability may be activated only once per turn and only on the controller's turn.
• CR 112.7a: An ability on the stack is independent of its source.