Can you respond to a Planeswalker's loyalty ability activation with a removal spell targeting that planeswalker?

When a player activates a planeswalker's loyalty ability, it goes on the stack just like any other activated ability (CR 606.1). After it is placed on the stack, players receive priority in turn order, giving each player a chance to respond before the ability resolves (CR 117.3b).

During this response window, you may cast a removal spell — such as a destroy, exile, or damage spell — that targets the planeswalker. If that removal spell resolves and destroys or exiles the planeswalker before the loyalty ability resolves, the planeswalker is gone from the battlefield (CR 606.3). However, the loyalty ability itself remains on the stack and will still resolve, because abilities exist independently of their source once placed on the stack (CR 112.7a).

This is a crucial distinction: killing the planeswalker in response does not counter or fizzle its loyalty ability. The ability will still resolve and its effect will happen (unless it requires a target that is now illegal). You prevent future use of the planeswalker's abilities, but not the current one already on the stack.

Example: Your opponent activates Liliana of the Veil's +1 loyalty ability (each player discards a card). In response, you cast Murder targeting Liliana. Murder resolves, Liliana dies — but her +1 ability is still on the stack and will resolve, forcing each player to discard a card.

If your removal spell targets the planeswalker itself and the planeswalker is no longer a legal target when your spell tries to resolve (already moved to graveyard by some other effect), your spell is countered by the game rules for having no legal targets (CR 608.2b). Plan accordingly.