When to answer and when not to answer a brief e-mail from a busy academic?

My own busy-person email triage has three tiers:

1. No response needed = near-zero cost
2. Minimal response needed = minimal cost
3. Response with significant thought and care needed = significant context-switching cost

A brief "thank you" email would go into category #1: no bother or burden, but noticed and appreciated all the same. In other words, send the email: it's not a significant nuisance, and it's nice to be appreciated.

To me, any email takes some amount of time to process. If I were to send something so short as "Can do. Best" (NB I wouldn't, but if I did) then I wouldn't be expecting a response, and in fact if I got one, to be honest, I would consider it a waste of time. Don't get me wrong, I'd appreciate the sentiment, and I wouldn't be offended, but I'd consider such an email wholly unnecessary and more distracting than it's worth. After all, what's the alternative - that you're not thankful? I think I can safely assume that, unless you're actually an all-around horrible person, that is not the case.

Contrasting this with jakebeal's answer, I think the only general conclusion you can draw is that depending on the person you're emailing, a "thanks" email in response to a short message of acknowledgement may be appreciated at best and somewhat inconvenient at worst, but it's unlikely to seriously bother anyone.

To me it depends on how well I know you.

1. If I know you very well, then we are probably exchanging e-mails frequently. In this case, the continuous streams of gratitude are a distraction and may even seem sycophantic.
2. If I know you somewhat well, and have already formed an opinion of you (presumably a good one, if I'm willing to do something for which you are thankful), then either way is fine, and everyone's preference will be different. Personally, I'd rather receive the "thanks!" mail only for really significant favors and omit them otherwise.
3. If I do not know you well, especially if our only contact was for you to request the favor, then I'd probably appreciate the "thanks!" mail, as I would have no other way of knowing that you appreciated the effort.

Given that you're requesting a LoR, you are probably in the second group, which is maybe why the answers are so mixed. In this case, my personal strategy would be to send a detailed "thank you" mail (or even a small gift) after all the letters are submitted, but I would not continuously thank them after every correspondence.