Thanissaro's translations are... unique. The word being used, as others have pointed out, is "mano", the leader of an irregular group of nouns called the "manogana" - "the group with mano as its leader".
The PED has a huge article on mano, worth reading in full if you're interested. I've copied the gist of it below, but as you should see, it is probably misleading to translate it as "heart". There is a word for that, "hadaya", which is sometimes used as a synonym of mano, viññāṇa, etc., but also refers to the physical heart organ.
It is fair to say he is within his right to use poetic license in this case; it is, after all, poetry, and the mind and the heart are much the same in Buddhism. As you can see below, though, manas generally refers more to the thinking side of things, than the emotive or the substantial entity of "the heart" or "the mind".
Mano & Mana(s); (nt.) [Vedic manaḥ, see etym. under maññati]
II. Meaning: mind, thought
- Mano represents the intellectual functioning of consciousness, while viñnāṇa represents the field of sense and sense -- reaction ("perception"), and citta the subjective aspect of consciousness (cp. Mrs. Rh. D. Buddhist Psychology p. 19) -- The rendering with "mind" covers most of the connotation; sometimes it may be translated "thought. As "mind" it embodies the rational faculty of man which, as the subjective side in our relation to the objective world, may be regarded as a special sense, acting on the world, a sense adapted to the rationality (reasonableness, dhamma) of the phenomena, as our eye is adapted to the visibility of the latter.
- As regards the relation of manas to citta, it may be stated that citta is more substantial (as indicated by translation "heart"), more elemental as the seat of emotion, whereas manas is the finer element, a subtler feeling or thinking as such. See also citta2 I., and on rel. to viññāṇa & citta see citta;2 IV. 2b. In the more popular opinion and general phraseology however manas is almost synonymous with citta as opposed to body, cittaŋ iti pi mano iti pi S ii.94.
As to your ancillary questions:
What is the original Pali phrasing?
As others have pointed out, the original is:
manopubbaṅgamā dhammā, manoseṭṭhā manomayā.
manasā ce paduṭṭhena, bhāsati vā karoti vā.
tato naṃ dukkhamanveti, cakkaṃva vahato padaṃ.
Which words are able to be translated freely between mind and heart?
Some words used more or less interchangeably are citta, viññāṇa, mano, hadaya, ceto and nāma. Any one of these could ostensibly be translated as either "heart" or "mind". The only one that literally means "heart", as mentioned, is hadaya. The rest have colloquial connotations and are derived from various roots, e.g. √cit = to think, √ñā = to know, √man = to consider, or not from roots at all (hadaya = heart, nāma = mentality).
What is the overall sense of the Pali words used i.e. what are the nuances or subtleties to the original Pali?
The technical meaning of the verse is that wholesome and unwholesome deeds come from wholesome or unwholesome mind states, not mere physical acts. In this, Thanissaro's translation is misleading. The emphasis should be on the fact that it is the mind as opposed to the body, not the heart as opposed to the mind.
The origin story about the elder Cakkhupala makes this clear; during his walking meditation, many insects were killed because he was blind. The Buddha spoke this verse to show that Cakkhupala was innocent.
To get an understanding of the colloquial meaning of this verse is a bit involved, but reading the PED's article on how mano is generally used might help. Its usage here does seem to be to emphasise the mental nature of guilt and innocence. Otherwise, citta is a more common word. The word for heart, hadaya, is used when talking about emotion, e.g. hadayaṅgamā - going to one's heart (i.e. heartwarming), and doesn't seem appropriate in this context.