This statement appears to be partially true.
The source of the claim is research conducted by Underwriters Laboratories. Underwriters Laboratories is approved by OSHA to perform safety testing, so they're likely at least somewhat trustworthy. It may be worth noting, however, that since 2012, Underwriters Laboratories has been part of the for-profit company UL LLC. The research that produced this claim was conducted in 2012.
The first thing that makes this claim only partially true is that furniture was only one variable in the experiments.
In 2012, scientists at UL (Underwriters Laboratory) designed a series of experiments that focused on the size and geometry of modern homes as well as current furnishings and building materials. The experiments tested three modern home configurations against three so-called “legacy” configurations containing furniture UL described as being similar to furniture made in the 1950s.
"Newer homes and furniture burn faster, giving you less time to escape a fire" - Today
"The backing of your carpet is synthetic, your drapes are synthetic, the couch, the pillows are synthetic," explained John Drengenberg, consumer safety director for UL. "They burn hotter and faster than natural materials do."
So, the claim's veracity could be easily improved - though perhaps only slightly - if it was changed to "...because of the flammability of modern home furnishings..."
It does not appear in-dispute that certain synthetic home furnishings can burn more quickly and hotter, when exposed to open flames; and it's likely the case that "newer" homes contain more of these kinds of furnishings.
However, the more we dig into this research, the shakier the broad conclusions seem.
We must note that the source of UL's claims is experimental data, not observations of real home fires.
One important thing to note about real house fires is that, according to "all available data from the 1970s through today, the vast majority of home fires that involve upholstered furniture are ignited by a smoldering source" (Time).
This is very important, because,
although “legacy” furniture made around the 1950s may ignite more slowly than “modern” furniture when exposed to an open flame, it ignites faster than “modern” furniture containing polyurethane foam when exposed to a smoldering source – which, again, is the most common ignition source in home fires involving upholstered furniture.
What this suggests is that, if you, for instance, drop a lit cigarette on your "modern" couch, it will take a long time to ignite, and may never do so. Whereas, if you dropped that cigarette on an chair with a hair and cotton seat pad, it might ignite quickly. After those fires have ignited, you might get different outcomes based on the kind of furnishings. But this raises the question:
You have 2-3 minutes to escape starting when? This question is proving strangely difficult to answer. Not only does UL not specify, many other sources give similarly vague warnings:
Ready.gov: "A fire can become life-threatening in just two minutes. A residence can be engulfed in flames in five minutes."
American Red Cross: "Two minutes is the amount of time that fire experts say you may have to safely escape a home fire before it’s too late."
SF-Fire.org: "In only 3 1/2 minutes, the heat from a house fire can reach over 1100 degrees Fahrenheit."
Building codes have been getting more strict and more sophisticated, and newer homes will have more and better placed smoke detectors, better wiring, and construction that's aimed to control the spread of fire.
FULL STATEMENT FROM THE NATIONAL ASSOCIATION OF HOME BUILDERS:
"According to the National Fire Protection Association, the leading causes of unintentional, nonconfined home fires are older heating and lighting equipment along with antiquated electrical distribution.
All data show that fatalities decrease when older, less safe homes are replaced with new homes that include safer construction based on newer building codes. These improvements include draft stopping in concealed spaces, safer appliances, changes to the electrical code and requiring hardwired, interconnected smoke alarms."