What happens to billy in the landlady?

Answer 1

Though one could imagine the ending of the Landlady due to the "magic" of cliffhanger endings, a more cynical and likely take on the story is the following:

One can assume that the old lady is a taxidermist (one who stuffs dead bodies) since she stuffed her parrot and dog, that she killed Mr. Mulholland and Mr. Temple, and that she stuffed them and kept them on the fourth floor. You can infer that Mr. Mulholland and Mr. Temple are dead because, near the end of the story, the old lady speaks of Mr. Mulholland in past tense:

"Mr. Mulholland was a great one for his tea. Never in my life have I seen anyone drink as much tea as dear, sweet Mr. Mulholland."

This implies one of two things:

1) Mr. Mulholland has left the house

2) Mr. Mulholland is dead

Since the old lady says that Mr. Mulholland and Mr. Temple are still on the fourth floor, you can imply that they are dead.

Additionally, Billy says that his tea tastes like bitter almonds. This is a characteristic of potassium cyanide (a very powerful poison). Given the above information, one can assume that Billy Weaver soon dies and is stuffed and kept with Mr. Mulholland and Mr. Temple on the fourth floor.

Answer 2

Billy gets poisend and stuffed just like all the pets in the landlady