The expression,"You lie like a dog in straw"

Early lying like a dog

The root expression seems to have been simply "lie like a dog." The earliest match for the phrase that an Early English Books Online search turns up is from Austin Saker, Narbonus: The Laberynth of Libertie (1580):

As for the Gentleman Courtier, his felicitie is not the happyest, nor his gaine the greatest: when he is in the prime of his youth, he must bee braue in apparell, and lauish of his pursse, neate in his going, and stately in his gate: courteous of behauiour, and curious in his choyce: liberall of his liuing, and no niggard of his loyaltie: his grace must be liked, and his maners marked: If he be a comely personage, then is hee hated of his inferiours: if faire, then effeminate: if blacke, then meeter to bee a Souldiour, then [fi]tte to be a Courteour: if well fauoured, disdeyned of the deformed: if kind and louing to all, then a Parasite, and a Flatterer: if something strange of acquayntance before he be knowen, then proude, and disdeyneth to speake: if riche, then a churle: if but of small reuenewes, then a shifter, and knoweth not elsewhere to liue: if high of stature, then a lubber: if low of making, then a Dwarfe: if liked of some Gentlewoman, then a hunter of that kinde: if he talke not in their companies, then precise, and tyed to his chastitie: if not content to put vp iniuries, a quareller, and a hacker: if patiente sometime to beare, rather than to make a brabbling for nothing, then is he a milkesop, and as good a man as Maulkin: if liked of the Nobilitie, and beloued of Gentlemen, then eyther he caries two faces in one hoode, or doth lye like a Dog: if disliked of a few not his freends, & them some vayne persons, therfore vnregarded of many, and not beloued of any: if he spend something liberally, then is he prodigall, and neuer mindeth to buy lands: if something hard, then miserable, and a churle: if liked of his Prince, then a faire tounged fellow, or he coulde neuer gayned such good will: if not loued for affection sake, then hated of all, and indéede rather a conspiratour, then of any honest demeanour.

The lying here appears to mean dissembling, although the sense is perhaps less clear than in the next example, where the meaning is unmistakable but then taken wrongly (as a jest) by the second party to the quoted dialogue. From John Taylor, A Dog of War, or, The Trauels of Drunkard, the Famous Curre of the Round-Woolstaple in Westminster His Seruices in the Netherlands, and Lately in France, with His Home Returne (1628):

But many pretty rediculous aspersions are cast vpon Dogges, so that it would make a Dogge laugh to heare and vnderstand them: As I haue heard a Man say, I am as hot as a Dogge, or, as cold as a Dogge; I sweate like a Dogge, (when indeed a Dogge neuer sweates,) as drunke as a Dogge, he swore like a Dogge: and one told a Man once, That his Wife was not to be beleeu'd, for she would lye like a Dog; marry (quoth the other) I would giue twelue pence to see that trick, for I haue seene a Dog to lye with his Nose in his Tayle.

From very early times, then, speakers were conscious of the competing meanings of lie as "tell an untruth" and "position oneself horizontally."

The earliest match for lying like a dog in something involves a manger, consistent with the information that Laurel provides in a comment above. From Thomas Dekker, The Honest Whore with, the Humours of the Patient Man, and the Longing Wife (1604):

Crambo. No more to doe, but insconce your selfe i'th taueren; prouide no great cheate, couple of Capons, some Phesants, Plouers, an Oringeado-pie or so: but how bloudy so ere the day be, sally you not forth.

Fustigo. No, no, nay if I stir, some body shal stinke: ile not budge: ile lie like a dog in a manger.

This usage seems to be intended literally as the behavior of a lazy dog; there is no hint here of dishonesty. To my surprise, I could not find another instance of "lying like a dog in a manger"—which would seem closely akin to "lying like a dog in straw"—after Dekker's until this instance in a speech by Representative Bertram Podell of New York titled "France: Faithless Ally and Betryer of Israel," in The Congressional Record (January 8, 1969):

She ["De Gaulle's France"] lies like a dog in the European manger, refusing entry into the Common Market to England, who bled for her, helped free her, and seeks to enter peacefully into a European economic union. It is perhaps the last best hope for a cure to the disease of European nationalism that has twice almost destroyed the world.

As in this instance, the familiar trope of "a dog in a manger" generally focuses on a very different type of behavior: denying something that is of no use to oneself to others who could benefit from it.

Meanwhile, "lying like a dog" in the "telling falsehoods" sense does appear in a variety of longer phrases, including "lie like a dog licking a plate" (sometimes rendered as "lie as fast as a dog will lick a plate"), "lie like a dog on a rug," "lie like a dog a-trotting," and "lie like a dog running on hot sand."

The association of lying (in the sense of telling falsehoods) with dogs evidently occurs in languages besides English. John Williams, The Rise and Progress of the Northern Governments; viz. the United Provinces, Denmark, Sweden, Russia, and Poland, volume 2 (1777) describes a judicial punishment against calumniators that, he says, is "still practiced in Poland":

When a calumniator is judicially convicted of his crime, he is conducted into the great hall of the senate, where he is obliged to lie down under the seat of him whom he has offended and there, in this humiliating situation, he is obliged to pronounce, with a loud voice, that he repents sincerely of the calumnies and falsehoods that he has wickedly spread against the reputation of the person whom he accused, and that he lied like a dog; and after this public confession the guilty person is obliged to counterfeit three times the barking of a dog, which terminates this singular scene.

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Lying like a dog in straw

I couldn't find any instances of this expression in newspaper or book databases online. The closest expression that I did find is from Walter De La Mere, "The Creatures," in The London Mercury (January 1920):

He stooped forward, lean, darkened, objurgatory. "Don't we make our world? Isn't that our blessed, our betrayed responsibility?"

I nodded, and ensconced myself, like a dog in straw, in the basest of all responses to a rare, even if eccentric, candour—caution.

The simile in this case is to a dog resting in a nest of straw. There is no sense that the narrator's nodding amounts to a falsehood, but that he is building a cocoon of caution around himself rather than responding to the other speaker's candor with openness of his own.

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Conclusion

I could not find any instances of "lie like a dog in straw" in the various online book and newspaper databases I checked. The expression may be used in pockets of the English-speaking world today, and its usage may go back a considerable distance into the past, but it hasn't left a trail that I could discover.

A brief scan with Ngram:

• lie like a dog in a manger
• lie like a dog in your bed
• lie like a dog in a kennel
• lie like a dog on a rug during a July day in Florida
• lie like a dog a-trotting
• lie like a dog licking a plate

There probably are more, but the string seems to pull up an awful lot of bad hits -- either the string is not found or the book text isn't accessible.

You seem to have conflated "you lie like a dog in straw", and "You lie like a dog" which mean different things.

A dog in straw is comfortable. Here “dog” is literal, and there is a punning use of “lie” - "You lie like a dog in straw" = “you tell untruths very comfortably” or “You sound very comfortable (i.e. 'at ease') when you say that.” The point here is that if you are telling untruths, your conscience should make you uncomfortable.

In “You lie like a dog”, there is no pun with "lie", which means "to tell untruths" and “dog” is used as a pejorative.