The various Smithsonian museums in Washington D.C. are always free (and federally funded!); once a year, Smithsonian magazine’s Museum Day twists the arms of other orgs to open their doors for free too, and this year it’s going to be September 24.
Some Texas art museums on board for the event include Crow Collection of Asian Art, Blanton Museum of Art, Dallas Museum of Art, Ellen Noel Art Museum in Odessa, Houston Center for Contemporary Craft, Houston Center for Photography, International Museum of Art & Science (IMAS) in McAllen, The Museum of Fine Arts, Houston, and the Tyler Museum of Art , and many others, some of which are free anyway! The rarity of museum free-ness is further spoiled by Houston Museum District Day on September 17, which also features many free admissions.