The Harp definitely has harmonics as well. There is a famous part of La Boheme that has a harp playing harmonics.
As a general rule, there are two things that are meant by saying the word harmonic. There is the botoon or overtone that is sometimes played by touching certain stringed instruments at certain places while playing.
More generally a harmonic is some tone that is played over some sort of base tone. Think of what a harmoniser pedal is on electric guitar, it is a pedal that takes the base tone of your guitar and plays another tone over it, a certain interval.
So bringing it back to your question can an Organ play harmonics? An organ may consist of a number of harmonics, it is not all that uncommon to multiple pipes blowing at a time and these pipes may all add a harmonic texture to the music.
Can a harp play harmonics, yes as the video demonstrates? I don't know enough of the working of brass instruments or flutes so I will leave those two for other answers.