I think "Iacke" here most likely refers literally to a rotating mechanism used to turn a spit (or perform other work by moving in a circular motion), which is used metaphorically as a symbol of a useful or productive process. "Censure will not [...] make the Iacke go" basically means "Censure isn't productive/doesn't do useful work/doesn't keep things running".
Such devices existed and were called jacks at the time the first folio was published (in 1623), as explained by the following note from a modern edition of "Shakespeare's Comedy of The Comedy of Errors", edited with notes by William J. Rolfe, 1904, page 131:
Machines or jacks for turning the spit, moved by weights like a clock, had been invented in the time of S. We find them mentioned as early as 1585 in the Nomenclator of Adrianus Junius: "automatarius faber, a maker of devises and motions that goe and turne of themselves, as clocks, jacks to turne spits," etc. In the preface to the folio of 1623, we read: "Censure will not driue a Trade, or make the Iacke go." In Brome's Antipodes, 1640, mention is made of a project "for putting downe the infinite use of jacks, whereby the education of young children, in turning spits, is greatly hindered."
See also the notes on jacks in Ancient Inventories of Furniture, Pictures, Tapestry. Plate, etc.
Here are two relevant Oxford English Dictionary definitions of jack, and a quotation showing that this word was sometimes used in metaphors:
A device for turning a spit for roasting meat over an open fire; = roasting-jack n. Now historical.
1564– II.7. Any of various devices consisting (solely or essentially) of a roller or winch; esp. a roller around which a towel is looped (English regional (midlands) in later use). Cf. sense II.11 and jack roller n. 1. Now rare.
1623 You should finde some Iacks faulty, and some cogges missing, whereby the wheele of Iustice is hindered in his circular course. Thomas Scott, Projector 26
Douglas Bruster (2013) ("The Representation Market of Early Modern England" Renaissance Drama, 41(1/2), 1–23, https://doi.org/10.1086/673903) agrees that definition II.7 is being used in the introduction to Shakespeare's first folio, saying
By “drive a Trade,” Shakespeare’s colleagues-cum-editors mean something like push, encourage, or sustain buying and selling in a particular field; “the Jack” refers to a wind-up mechanism, or parts controlled by that mechanism, such as a spit that holds and turns roasting meat, or the still-familiar human figures that strike bells on the outside of clocks.
(for continuation of this quotation, see footnote 1 below.)
Here are other quotations I found on Google Books that seemed relevant to me:
The bridle governs the horse, the spur quickens, him; the weight upon the line makes the Jack go; the oil upon the wheel makes it go glib, and nimble: The sayls give the speed, the ballast steadiness to the motion of the shhip. And hereupon God weighs out to us our favours and crosses in an equall balance; and so tempers our sorrows, that they may not oppress; and our joys, that they may not transport us.
(Richard Younge, 1655, A Christian Library; or, a pleasant and plentiful paradise of practical divinity..., London, pages 216-217)
Based on the sense of the passage, we can infer that the "Jack" mentioned is something that has a line that can be weighted and a wheel that can be oiled.
Well good Landlady set on the pot, as they say; or make the Jack go; then I'll hear you.
(The Wild Gallant, John Dryden, 1684, page 5)
By the way, this word originates from the name "Jack" and would have been pronounced the same way in Shakespeare's time, as one syllable, close to the modern pronunciation of "Jack" (i.e. like /dʒak/ or /dʒæk/): the letter I was used at the time as a spelling of the consonant sound /dʒ/, and the convention of using E as a "silent" letter at the end of words was already established (although in certain contexts, such as words ending in "-ed", a syllabic pronunciation of the letter E was more common than it is today).
This question is difficult since the word jack/Jack has so many senses. I'm not completely certain of my explanation, but it is supported explicitly by Rolfe (1904) and Bruster (2013), and I think the quotations from Thomas Scott (1623) and Richard Younge (1655) also support it by showing that mechanical jacks were familiar enough to be used as metaphors at the time when the First Folio was published. So this definition currently makes more sense to me than interpreting "Iacke" as a person in this context (an intepretation that can clearly be ruled out in the case of the similarly worded quotation from Younge).
Bolding in quotations was added by me.
- Bruster 2013's explanation is as follows:
See Oxford English Dictionary, s.v. “drive,” verb, def. IV.19, and “Jack” noun1, def. II.7, quoting Thomas Dekker, who, in The Seven Deadly Sins (1606), deploys it alongside reference to a clock mechanism: “It stood altogether like a Germane clock, or an English Iack or Turne-spit, vpon skrewes and vices.” Also relevant is sense 6, here, reflected in Richard III’s memorable censure of Buckingham: “like a Jack, thou keep’st the stroke / Betwixt thy begging and my meditation” (4.2.117). The implication of these phrases is clear: in any trade, things “drive” (are driven) and “go” only when customers buy.
Sense 6 in the OED refers to another kind of mechanical device:
1498– II.6. (A name for) a mechanical figure of a man which strikes a bell on a clock or clock tower at certain times; = Jack of the clock n. at Phrases P.3. Now chiefly historical.