In the book Number the Stars, why does mama tell the soldiers that they need to keep casket closed because an aunt died of infectious disease?

Mama says that Birte died of an infectious disease so that no one will look in the casket since Birte is not in there.

Great Aunt Birte does not exist.  She is an invention.  When Annemarie finds out from Uncle Hendrik that her aunt has died, she is a bit confused.  Henrik makes a speech about why they need a coffin in the house.

“There has been a death, and tonight your Great-aunt Birte will be resting in the living room, in her casket, before she is buried tomorrow. It is the old custom, you know, for the dead to rest at home, and their loved ones to be with them before burial." (Ch. 8)

Annemarie knows that something is going on because Great Aunt Birte is a fiction.  She is old enough to be curious about what is really going on.  She confronts her uncle, because her mama and uncle are lying to her and she wants to know why.  He tells her that the less she knows, the braver she can be.

Part of the custom is for people to gather at the house of the dead person’s loved ones.  However, this meeting is not unobserved by the Nazis.  The “death” in the family is an excuse for people to be gathered.  Mama says the casket is closed because Birte died of typhus germs, pretty much guaranteeing no one will open it to check their story.  It works.  The Nazi who questions them refuses to let her open the casket.

"You foolish woman, “he spat. "To think that we have any interest in seeing the body of your diseased aunt! Open it after we leave, “he said. (Ch. 10)

When the casket is opened later after the Nazis have gone, it is actually “stuffed with folded blankets and articles of clothing” (Ch. 11) to be distributed to the Jews in hiding in the house pretending to be mourners.  Annemarie soon learns that her family is part of the Resistance, and the story about Birte is just a cover to get the Jews to safety out of the country.

Annemarie's uncle had asked her if she could be brave, and told her that she would be braver if she did not know everything.  Annemarie finally understands with this incident.  She has to grow up quickly, and demonstrates bravery and maturity in telling the Nazi that her great aunt is in the coffin, and in taking the secret package to her uncle.  She knows just enough about what is going on to be scared, but still plays her part.