A rug lies on the floor. Someone who “lies like a rug” is someone who always lies (or prevaricates). The saying is frequently applied to politicians who don’t tell the truth.
 
“You lie like a rug” was cited in print since at least 1923.

 
Merriam-Webster Dictionary
lie like a rug
US informal
:  to lie constantly :  to be dishonest
     
Google Books
Gargoyle Magazine
Volume 17
1923
Pg. ?:
YOU LIE LIKE A RUG!
 
12 March 1929, Green Bay (WI) Press-Gazette, “Staff Chaff,” pg. 6, col. 3:
The high school boys are studying figures of speech this term. “You’re a liar,” I heard one of them say yesterday; “you lie like a rug.”
 
18 February 1937,  San Luis Obispo (CA) Daily Telegram, “Trib-o-Grams,” pg. 4, col. 3:
Gashouse Gus—Don’t you believe me?
Joe Bungstarter—No, you lie like a rug.
   
Google Books
Body, Boots, & Britches:
Folktales, Ballads, and Speech from Country New York
By Harold William Thompson
Philadelphia, PA: Lippincott
1939
Pg. 497:
Lie like a rug
 
Google Books
The Synonym Finder
Edited by Jerome Irving Rodale
Emmaus, PA: Rodale Press, Inc.
1978
Pg. 562:
... lie like a rug; ...
 
OCLC WorldCat record
He Lies like a Rug: Digitising Memory
Author: Tara Brabazon
Edition/Format: Article Article
Publication: Media International Australia, v96 n1 (8/2000): 153-161
Database: SAGE Journals
Summary:
With most critical and historical interpretations of the Internet trapped between the ideologies of technophilia and the progress, little attention is being placed on the digital future of the analogue past. This paper evaluates contemporary notions of cataloguing, preservation and memory. However, most stress is placed on how the passions and enthusiasm of popular culture are tempered through digitisation.
 
OCLC WorldCat record
Lie like a rug
Author: Donna Huston Murray
Publisher: New York : St. Martin’s Paperbacks, 2001.
Edition/Format:   Print book : Fiction : English
 
Urban Dictionary
lie like a rug
To lie (tell a falsehood); to lie through one’s teeth. A pun on lying down, as a rug does.
“Hide the cookies under the crib and lie like a rug,” Binky whispered to Charlie as they heard their parents returning home.
#lie through one’s teeth #lie #cheat #fib #whopper #deceive
by knightshade May 07, 2006
   
Twitter
Tim Black ™
‏@RealTimBlack
Hillary says she has “Public positions and Private positions” on issues. Translation: She’s a con artist and lies like a rug when needed.
9:49 PM - 8 Oct 2016
 
Twitter
Michial Brown
‏@greetandtweet
@thetruthranger @FoxNews @clairecmc @realDonaldTrump Hardly. Trump is brutally honest. No liar. Was ill mannered. Hillary lies like a rug.
4:05 PM - 9 Oct 2016