In a panning shot of the Manhattan skyline we catch a glimpse of the PanAm (Metlife) Building and the Metlife sign. In the MCU (of which Iron Fist is a part) that building is the Avenger's Tower.
At 8:40 in Ward's office Danny leaves his backpack (tries to pick it up but security does not let him), but at 15:40 he is again shown with it, although he did not have it while entering Joy's apartment.
Danny Rand quotes Buddha as saying: "Your purpose in life is to find your purpose". However, the full quote is: "Your purpose in life is to find your purpose and give your whole heart and soul to it," and although it circulates widely in social media, it is a fake Buddha quote. It's a variant of "Your work is to discover your world and then with all your heart give yourself to it."
The practice swords used for the martial art known as Kendo are shown in Colleen Wing's dojo (gym). They are called "shinai" and are very flexible, with a bamboo core. When new - before use - and for shipping, a shinai is wrapped with bands. The shinai shown in the show still have those bands in place. Also, the martial (colored) belts closing the kimonos of Colleen's pupils are badly done, like a true beginner would.
In episode 4, the tattoo on Danny Rand's chest is seen to extend to a height between his armpit and the top of his shoulder, and the tip of the dragon's right wing should consequently have been visible in the open collar of his shirt, which in some scenes in episode 1 opens to a point even lower than his armpit.
In one scene, Chinese New Year (February 16th) is being celebrated in the streets of New York. February in NYC is bitter cold; all the extras are dressed in Summer clothing. Ironically, the title is "Snow Gives Way (#1.1)."
When the guy was searching on his iPhone, the time appears to be 10 AM, while you can clearly see that it is nighttime.