If your teacher says your position on the benefits of abolishing homework is not tenable, she means it is not based in fact. To be tenable is to be evidence-based and well-founded.
Tenable comes from the Latin root tenir which means "to hold," as in "hold together." If your plan is tenable, it will probably hold together when you execute it, or hold up to scrutiny. It's good to note that the opposite of tenable is untenable and not intenable, though throughout history both have been used.