In episode 126 of Gintama, Shimura Shinpachi was seen supervising the Terakado Tsuu's Imperial Guards dance training. However, seeing μ's concert, none of the fans do such dance in both the anime and in the live concert.
Where does the dance origin from and does it have a name?
The dance is called Wotagei or Otagei. It is a dance done by Idol Otaku (called Wota) to show support for their idol during the concert. Quoting Wikipedia,
Wotagei is particularly associated with fans of Hello! Project and AKB48 idols, as well as fans of anime voice actresses, who often perform theme songs for the series in which they appear.
Thus, it can be said the the dance Terakado Tsuu Imperial Guard was practicing is originated from the real world wota of AKB48 and other idol groups. I never watched idol concert other than the μ's one, but such dance is possibly done often by AKB48 wotas, considering that a member of AKB48, Sashihara Rino, has done a mimic of wota dance during one of AKBingo show, suggesting that such a thing is common during AKB48's concert. This is further reinforced by the fact that each idol group, and possibly each wota group has their own dance and chant pattern as shown here and here. Morning Musume wota and AKB48 wota has their own chanting pattern.
As for why μ's wota are not doing wotagei during their concert, it might be due to the seating disabling them from moving too much. It might also be because among otaku culture itself there are categories of otaku, such as Anime Otaku, Idol Otaku, Game Otaku, etc. Since μ's comes from anime, the fans attending the concert would be a mix of different types of otaku, thus, the wotas might refrain themselves from doing such a dance so that they won't bother the other fans. This might also be a rule put by the concert management.