An affiche is a poster (of a show, a movie, a play), so the idiom

tenir l’affiche

means a show is “holding its poster”, i.e. is currently running somewhere and has enough success to keep going day after day.

There are also the expressions:

être à l’affiche

which just means a show is running now, without giving any indication about its success and

tenir/occuper/partager/quitter/… le haut de l’affiche (“to hold/occupy/share/leave/… the top of the poster”)

which means that a show or a part of a show is the most successful of its kind, or a person has his/her name listed first on the poster of a show/movie, or is simply one of the leaders in some domain.