On June 23, 2014, President Barack Obama stopped at a Chipotle Mexican Grill in the Woodley Park area of Washington, D.C., for a takeout lunch (widely reported around the White House Summit on Working Families).

Contemporaneous news coverage and photos establish two core points:

• He ordered at the counter. Outlets including TODAY, Eater, E! Online, and the Los Angeles Times described him building an order at Chipotle. Chipotle's national office told press the order looked like a chicken burrito bowl (not a wrapped burrito). The viral framing often says "burrito" loosely; the documented order type was a bowl.
• He leaned over the sneeze guard and pointed at ingredients. White House photographer Pete Souza posted a photo of Obama with his head and arm past the glass barrier, index finger extended toward the line ingredients. Multiple outlets republished the image and described the same pose. Social media and TV anchors labeled the moment a hygiene/etiquette "faux pas" ("SneezeGate"). Chipotle's office was quoted saying, in substance, that sneeze-guard rules still applied and he should not have reached over.

The photograph is primary visual evidence of the lean-and-point. It was distributed by the official White House photographer and by wire photographers (e.g. AFP/Getty frames from the same stop also circulated). Newsrooms that covered the story treated the image as authentic, not a composite or later edit.

In plain language: Yes. In June 2014 Obama ordered a Chipotle chicken burrito bowl in D.C. and was photographed leaning over the sneeze guard while pointing at toppings. The “burrito” wording is slightly loose (bowl, not wrap); the sneeze-guard lean and pointing are well documented.