In the snippet "The teacher's questions didn’t __ any answers from the class," which word best fits the blank?

There's only one word that fits. To see this think about the standard meaning of each word:

'Elicit' means to call forth or draw out as information or a response. For example:

'her remarks elicited cheers'; 'the survey elicited the views of the firm's clients'

'Illicit' means unlawful or illegal.

'Elate' means to make (someone) ecstatically happy.

So, filling in the blank we have: "The teacher's questions failed to elicit any answers from the class."

'Allocute' means to make an allocution or to question a criminal defendant prior to sentencing.