"Warm the affections" is a phrase found in many religious texts going back to the early eighteenth century (at least), and I think it was then and continues to be used in a sense that is nowadays somewhat archaic.

"And consequently, tho' extempore Prayers may warm the Affections; yet 'tis impossible for the Congregation to discharge that Warmth aster such a manner as is requisite in prayer"
Thomas Bennet (1708)
https://books.google.es/books?id=3FwdDT88Yq4C&pg=RA1-PA78

"Knowledge alone is like a winter-sun, which hath no heat or influence ; it doth not warm the affections, or purify the conscience."
https://books.google.es/books?id=ZsoHAAAAQAAJ&pg=PA643&lpg=P...

a. The introduction or exordium of a sermon
1) The various functions of the introduction
[...]
c) It should warm the affections or emotions of our hearers to the subject or substance of the sermon."
http://www.beingtaught.org/lecture-4-constituent-elements-in...


"Affection" nowadays usually means fondness, liking, even love. It can be pluralised: you can talk about "capturing someone's affections" or winning a place in someone's affections", meaning make them fond of you.
https://en.oxforddictionaries.com/definition/affection

However, "affections" used to, and sometimes still does, mean mental states, emotions or passions of any kind. In 1755, Dr Johnson defined "affection" first as "the state of being affected", second as "passion of any kind", and third as "love, kindness, goodwill to some persons". The third use is the modern one, and is usually singular. The second is the relevant one here (I believe) and is often plural. Johnson illustrates it with this passage from Hooker:

"Affections, as joy, gried, fear, and anger..."
https://archive.org/stream/dictionaryofengl01johnuoft#page/n...

In other words, "affections" here means the same as "afectos" in Spanish, though that's not its usual meaning in modern English.

As for "warm", I think it clearly means arouse, stimulate or engage. Johnson again:

"To heat mentally; to make vehement"
https://archive.org/stream/dictionaryofengl02johnuoft#page/n...
And that, I think, is clearly what it means in the passages quoted and in your text.

This expresses the idea that in prayer or preaching cold, emotionless knowledge is useless unless the emotions are engaged. These emotions are not confined to "affection" in the modern sense of fondness. If the subject is sin or evil, a vehement repudiation is called for, more than just a rational knowledge that it is to be rejected.

--------------------------------------------------
Note added at 3 hrs (2018-06-25 10:50:23 GMT)
--------------------------------------------------

Contemporary religious discourse very often uses expressions with a long history, inherited from older texts: archaic expressions. I think this is an example of that.

--------------------------------------------------
Note added at 3 hrs (2018-06-25 10:52:24 GMT)
--------------------------------------------------

My second quotation is from a text of 1837 by Thomas Watson.