This ties in to the recent question: Why "Would you mind if I asked you something?" is correct?

What is going on here is that we have a conditional "if" on a clause which is based on the modal verb combination "will fall". Since this is in an irrealis mood, the verb goes into the subjunctive form. And because the other clause ("other countries ... would fall ...") is in the past, it also goes into the past to agree with that clause.

The usage has somewhat of an archaic ring to it, similar to uses like "Oh if I would be rich", or "Would that he be master". In general, there used to be more flexibility in the language with the verbs "shall", and "will" and their past participles.

Through modern eyes, the sentence looks as if it is derived from the structure "if X will [verb], Y will [verb]" which is not used very much in English. When we relate two future events together in modern English, we use the present tense for one of them (in English, present tense denotes future actions in some contexts like this one!). So the canonical sentence is "If X [verb]s, Y will [verb]":

If you eat my candies, I will be upset.

* If you will eat my candies, I will be upset.

Since the will-will structure is odd, so is the would-would structure:

* If you would eat my candies, I would be upset.

So the preferred canonical sentence is:

If Vietnam falls, other countries in South Asia will fall too.

And from this form we can derive other tenses. For instance, we can hoist it into the irrealis mood without changing the meaning, by changing both clauses to the past subjunctive form:

If Vietnam fell, other countries in South Asia would fall too.

_{Note that the past subjunctive is the same as the past tense for "to fall": namely "fell". But for other verbs like "to be", it changes. For instance:}

^{Indicative: If Vietnam is conquered ...}
_{Irrealis: If Vietnam were conquered ... [Subjunctive past of "to be" = "were"].}

Further, we can derive more tenses from our basic sentence:

If Vietnam had fallen, other countries in South Asia would have fallen too.

If Vietnam is falling, other countries in South Asia will be falling* too.

There is another way to express this, namely the "be + to [verb-infinitive]" form. The basic sentence for that is:

If Vietnam is to fall, other countries in South Asia will fall.

This form is not used. Speakers instinctively transform it into the past subjunctive, resulting in this very common form:

If Vietnam were to fall, other countries in South Asia would fall.

This "be + to [vinf]" form is quite restrictive in the formation of tense. There is no "will be to fall" or "would be to fall" or "have been to fall". There is "was to fall", which is a special form that has a meaning like "was expected to fall":

He was to finish the project by the 15th, but it overran right to the end of the month.

So to recap, the original sentence is a little archaic since it uses a would-clause as a conditional:

If Vietnam would fall, other countries in Southeast Asia would fall too.

It can be understood as being an older form of these modern usages:

If Vietnam {fell | were to fall}, other countries in Southeast Asia would fall too. [Subjunctive.]

If Vietnam falls, other countries in Southeast Asia will fall too. [Indicative.]

We can understand it as simply conditionally relating two clauses, which are expressed as a past subjunctive form of "to will": a subjunctive, irrealis form of the sentence "If Vietnam will fall, other countries will fall" whose meaning is easy to understand, but which is somewhat grammatically odd to the modern ear.