Can the landlord ask me to watch the neighbors for her?
I've been asked to watch the neighbors because our landlord thinks they will move out in the middle of the night without paying what they owe. Is this something the landlord can request of me?
No it is totally against ethics and law. You can not spy on anyone. And if your landlord is saying for this then ask him why would I do it? It is not your job to abide by his dictations. Say him clearly that you can not do this.
The landlord can ask--you don't have to accept. If a condition is placed on the request--"I need you to watch them or I'll have to evict you", then there's a problem. It's unrealistic to expect you to watch all the time--it's not your job, it's his. But if you notice them moving, why not let him know? It never hurts to have your landlord appreciate you (he'll remember it if you're a few days late with the rent, maybe).