When should I write a thank you letter to the author of a scholarly paper?
It certainly wouldn't hurt, and as a published author myself, I would say it would be very nice feedback, particularly if you include details about:


specifically any particular points/methods you found useful.
A brief outline of how you are extending the work.
Perhaps any question you have about the article.


This could be a good way to make yourself known in Academia, especially if the author is an influential scholar in your field.  However, a caveat, don't overdo it - in terms of being 'over the top' complementary or too many 'rapid fire' letters/emails.

But, it would more than likely be appreciated as it would be a validation of the author's work.