What does plaster in "plaster saint" refer to?

The point is that a plaster saint can't actually do anything, it's just an object that's supposed to remind one of the real person, but has no powers of its own. Saints are saints generally because they are supposed to have caused miracles to be performed. Plaster saints do nothing at all but sit and stare at you from their niches in churches.

So you are no plaster saint means you don't just look like a saint, you're the real thing. It's a real compliment.

The interesting thing is Merriam-Webster defines plaster saint simply as:

a person without human failings.

I sifted through Google Books, and this is the meaning you find in book after book after book. When it is explained why someone is not a plaster saint, the reason is that the person is less than saintly, misbehaves, has passions, struggles with temptation, very much unlike the other-worldly, beatific, ideal represented by a plaster saint, or the lifeless object itself.

It’s not hard to imagine how plaster saint could come to mean hypocrite: real humans are flawed; if you look like a plaster saint you must be faking it. Sarcasm could have played a role here too. However Bernard Shaw’s quote is highly atypical. Annie Edwards’ A Plaster Saint (1899) is the only other instance I found of plaster saint used sarcastically in this way.

Plaster saint in the Merriam-Webster sense appears in scores of books though. Merriam-Webster says the first known use is in 1890. So it’s likely Kipling’s Tommy, quoted by the OP, and first published that year under the title The Queen’s Uniform (The Kipling Society). Here Kipling has the proverbial British soldier Tommy Atkins criticise the British public, who sees the common soldier sometimes as a hero, sometimes as a ruffian (my emphasis throughout):

[…] Yes, makin’ mock o’ uniforms that guard you while you sleep
Is cheaper than them uniforms, an’ they’re starvation cheap.
An’ hustlin’ drunken soldiers when they’re goin’ large a bit
Is five times better business than paradin’ in full kit.
Then it’s Tommy this, an’ Tommy that, an’ Tommy, ’ow’s yer soul?
But it’s “Thin red line of ’eroes” when the drums begin to roll
The drums begin to roll, my boys, the drums begin to roll,
O it's “Thin red line of ’eroes,” when the drums begin to roll.

We aren’t no thin red ’eroes, nor we aren’t no blackguards too,
But single men in barricks, most remarkable like you;
An’ if sometimes our conduck isn’t all your fancy paints,
Why, single men in barricks don’t grow into plaster saints […]
Rudyard Kipling, Tommy aka The Queen’s Uniform (1890), (more info in Kipling Society) and full poem here

The following give a more explicit description of what a plaster saint is not:

Henry Morgan the Buccaneer was no “plaster saint”. His weaknesses, his follies, his errors are writ large on his record. He was rash, impulsive, reckless of speech, and oftentimes unscrupulous in action. He was a good hater and a firm friend.
The Transactions of the Honourable Society of Cymmrodorion, 1899, p. 41.

A study of his career will probably make us like him better, for we shall find that he was a man with very human virtues and failings, not a preposterous plaster saint.
William Alfred Hirst, Walks about London, Henry Holt, 1900, p. 80.

Sometimes the plaster saint is implicitly presented as something good:

“Look here, Elizabeth,” she said desperately, “have done with all this nonsense, for heaven's sake, and take your husband as you find him. He is no plaster saint, but neither are you, or any of us for that matter.”
Kate Horn, Ships of Desire, Cassel and Company, 1909, p. 317.

Sometimes people cultivate a plaster-saint image of important persons:

In short, she [Rose Parks] is on her way to becoming the secular version of a plaster saint. It is a fate that has already befallen Martin Luther King, who is so venerated it is politically incorrect even to acknowledge his human failings, like his womanising and his plagiarism.
“American trouble-makers,” The Economist Year Book, 1992 in Review, The Economist Books, 1993, p. 292.

the Trustees were aware of the existence of letters by Einstein, some of them since published, 15 others to be published later, that conflict with the “plaster saint” image they wished to preserve
John Stachel, Eistein from B to Z, Birkhäuser, 2002, p. 99.

Several strategies combine to defuse the image of Lincoln as a plaster saint. […] He may have contracted syphilis as a young man. […] Vidal’s very human Lincoln knows the art of the political deal.
Susan Baker, Curtis S. Gibson, Gore Vidal: A Critical Companion, Greenwood Press, 1997, p. 88.

Often we get the notion that a plaster saint is someone we wouldn’t like that much. In this case Ernie, a young girl, has a “mercurial temperament”, but is urged by her friends to beat another girl, this one perhaps a plaster saint by nature or conviction, at winning a prize for “the pupil whose general average in attendance, conduct, and scholarship should be the highest.” She says:

”All right,” promised Ernie, with a weary little sigh. “I don't mind the studying so much; but I must confess I'm tired of being a plaster saint!”
Alice Calhoun Haines, The Luck of the Dudley Grahams, Henry Holt, 1907, p. 173. (Full book available here.)

And even real saints are no plaster saints

Do not for one moment picture him [Saint John Bosco] as a little monster of perfection, with no personality, no reactions, anaemic as a plaster saint. The retiring, timid, peaceable, passive one was not John, but his brother Joseph—an intelligent, hardworking boy, marked from the beginning with mark of those who will never go above or below the level of a decent obscurity. But John was a different matter […]
Henry Ghéon, The Secret of Saint John Bosco, Tradibooks, 1944, p. 21.

As to whether plaster saint was borrowed from another language I found no evidence of. The literal equivalents plâtre saint or saint de plâtre in French, santo de yesso in Spanish, and santo de gesso in Portuguese, mean the object only. Petit saint (little saint) is used ironically to refer to a person hypocritically affecting virtue, and has been in use since the early 1800s. Ce n’est pas un (petit) saint (he/she is no (little) saint) sounds very much like the English phrase and has been around since the early 1800s too, but means they’re dishonest. My hunch, for all it’s worth, is that the phrase as used in the examples above is transparent and suggestive enough for English speakers to have coined it without outside help.

For what it's worth, I always thought it was a comparison with the statues of saints carved in marble that you would find in a cathedral. The plaster saint is a cheap, breakable imitation of the real thing.

Just speculation on my part.

I have always seen a plaster statue of a saint as an object of veneration and respect,

Unfortunately, the rest of UK had not had the same experience:

Some background… up until the late 18th century, “Plaster saints” were associated with Catholics who were seen as dangerous, weird, and enemies of civilisation as their allegiance was to the Pope and not the Crown.

In 1828, Daniel O’Connor, an Irish Catholic, was elected to Parliament but refused to take his seat until the anti-catholic oath was altered to his liking.

From The Encyclopaedia Britannica

O’Connell’s ensuing triumphant election compelled the British prime minister, the Duke of Wellington, and Sir Robert Peel to carry the Emancipation Act of 1829 in Parliament. This act admitted Irish and English Roman Catholics to Parliament and to all but a handful of public offices. With the Universities Tests Act of 1871, which opened the universities to Roman Catholics, Catholic Emancipation in the United Kingdom was virtually complete.

You will see that at the time of Kipling, Roman Catholics had only recently been (almost*) fully emancipated, and the cynical common soldiery, had not quite agreed to this – they, and the majority of the UK population saw the plaster saints that Catholics worshipped, and the Protestants did not, as tacky, and idolatrous – a cheap commercial representation of someone who, in fact, should be “an object of veneration and respect.”

Hence the derogatory use.

The OED gives:

plaster saint n. freq. derogatory a person who makes a show of being without moral faults or human weakness, esp. in a hypocritical way.

As has been shown, the first use must have been somewhat prior to Kipling’s 1890 use: “Single men in barricks [sic] don't grow into plaster saints.” and seems to indicate “replicas of truly holy saints.” This seems to be borne out by

1980 Chinweizu et al. in D. Walder Lit. in Mod. World (1990) 286 Were our ancestors a parade of plaster saints who never, among themselves, struck a blow or hurt a fly?

1995 Denver Post 15 Jan. e8/2 Clarke's book..presents her as a profoundly complex human being, infinitely more fascinating than any plaster saint or media-manufactured martyr.

There is also a play “A Plaster Saint” by Annie Edwards and what I have read of the context seems to accord with the derogatory use. “Historical Plays, Parts 1-7" By Tom Taylor from 1877 makes mention of it.

*There are still one or two restrictions on the rights of Catholics in the UK.