If you grow a large quantity of plants indoors, using either soil based sub-irrigation or clay pebble based hydroculture (particularly if fan-assisted) there may be some benefit regarding air filtration of volatile organic compounds (VOCs).
If you grow houseplants in drain hole pots in the traditional (but incorrect) way, any air filtration benefit is dubious to doubtful. Claims to the contrary are in the category of junk science.
Watch the following videos and you will see that this does not stop those with a vested interest in promoting it as good science.
The broad term "NASA research" giving implied endorsement of houseplant air cleaning means primarily a now retired NASA scientist by the name of Bill Wolverton. The objective of his testing was to explore growing plants in space ships. He did his testing in enclosed test chambers not open rooms as we have in our homes and offices.
The interior plantscaping and houseplant industries discovered Wolverton’s research and are primarily responsible for blowing this research all out of proportion. They seized on it and established the Plants for Clean Air Council, a marketing initiative. We know what the media is capable of doing with buzz-worthy information like this and they did.
Aided and abetted by the notoriety he received, Bill Wolverton went on to write a houseplant book titled How to Grow Fresh Air: 50 Houseplants that Purify Your Home or Office. Read this prior post What Did Bill Really Say? Incidentally, the book is now ranked No. 217 in the Amazon “houseplant” category. Fifteen minutes of fame has obviously passed.
What does the Environmental Protection Agency (EPA) have to say about houseplants and pollutant removal?
From An Introduction to Indoor Air Quality (IAQ)
Over the past few years, there has been some publicity suggesting that houseplants have been shown to reduce levels of some chemicals in laboratory experiments. There is currently no evidence, however, that a reasonable number of houseplants remove significant quantities of pollutants in homes and offices.
Indoor houseplants should not be over-watered because overly damp soil may promote the growth of microorganisms which can affect allergic individuals.
Also read a PDF by the EPA, entitled Care for Your Air: A Guide to Indoor Air Quality. A search will reveal no hits on either “plant” or “houseplant” (singular or plural).
Once again, the Latin admonition caveat emptor "Let the buyer beware", prevails in the consumer horticulture field.