> My English teacher is driving me nuts with the way she reads poetry. She says that at the end of a line without punctuation, you should just barge on ahead to the next line without a pause unless there's a natural pause there.

That's universally correct.

In my opinion, more than reading poetry your question is about interpreting a poem; the pause between two lines may depend on the text.

Many of the examples I have in mind are related to Italian poetry (from Dante Alighieri to modern days). Other examples may come from Whitman and Edgar Lee Masters ("Spoon River antology") but I don't remember exact quotes right now.

(1)
The Italian text of poems by G.Ungaretti can be understood only by pausing after each line (but this comes natural only when the reader understands why Ungaretti wrote those poems); reading them without pausing is totally meaningless.

(2)
Reading Dante Alighieri rhymes without pausing is pretty impossible (the metric is quite complex and latin poetry has 10 or 11 syllabes per line)

(3)
Paul Valery's "Le Cimetiére Marin" is the most intriguing example I know about.
Let's see the first lines:

Ce toit tranquille, où marchent des colombes,
Entre les pins palpite, entre les tombes;
Midi le juste y compose de feux
La mer, la mer, toujours recommencée
O récompense après une pensée
Qu'un long regard sur le calme des dieux!

The first two have punctuation; but this text starts as a "Moderato" and goes on as an "Andante"
Pause after "colombes", pause a bit more after "tombes"; then, the sun sets a fire in the audience and waves on the seashore give the rythm to the 3rd and 4th line in an "Accelerando", slowing down on "le calme des dieux". Long pause. Take a breath.
Here, pausing after each line would break the rythm, would throw away the pathos.

Go on.

Zénon! Cruel Zénon! Zénon d'Êlée!
M'as-tu percé de cette flèche ailée
Qui vibre, vole, et qui ne vole pas!
Le son m'enfante et la flèche me tue!
Ah! le soleil . . . Quelle ombre de tortue
Pour l'âme, Achille immobile à grands pas!

"Andante con brio", our eyes are following a flying dart. Don't pause after "aileè" or you'll break its flight! It vibrates, flies, and suddenly stops! Pause, because the arrow hit us, our soul. Pause.

Just for the notice, despite the fact that French and Italian are very similar and poetry has about the same metric in the two languages, translating the original text of Paul Valery's poem to Italian has been one of the most difficult and intriguing tasks of contemporary poetry. The first edition of the "Cimetiére" dates back to 1920, the first universally approved Italian translation was published not before 1963.