I think the past is used for the man simply because he’s dead. You can’t say “Il vient de boire du thé” talking about a dead man…

You’ve got to use the past tense for dead people.

I believe you are incorrect in your interpretation in the order of events. The spouse served tea twice : once for the alleged murder in the past, and once for the narrator in the more recent past (narrated as present).

Il a dû être empoisonné.

“He must have been poisoned.” : the man is currently dead, but it happened before. The narrator makes an hypothesis about the cause : poison.

Il venait de boire du thé, non ?

“He had just been drinking tea, hadn’t he?” : just before dying, the man had drunk tea. The “past before past” (passé antérieur) explicits this.
The stress in the final “, non?” creates a tension with the next sentence : the narrator implies that there will be a second part (a conclusion maybe) to this sentence.

Son épouse vient tout juste de nous servir du thé !

“His spouse has just served us some tea!”. The narrator now exposes the expected second part of the previous sentence. The narrator (“nous”) has been served tea very recently, during an event separate from the murder story, and correlates it with the hypothesis of poisonning.
The use of the Present is appropriate because the event of the narrator being served tea is still very close in time, and it also clearly shows that this is in a separate time from the murder story.