Rimmer's monkey painting was an ornamental, gold-framed painting owned by Arnold Rimmer, and was one of his most prized possessions.
When Rimmer created a hologram clone of himself, he decided to move in with his doppelganger into a new bunkroom, and leave Dave Lister alone in their old quarters. Since Rimmer couldn't move it himself, he had Lister bring the painting to the double Rimmer bunkroom, asking Lister to be extra careful with it in transit, since it was "priceless". ("Me²", Series I)
Rimmer had a new monkey picture brought to his cell in the Tank, although this was slightly different to the original painting (perhaps recreated differently by the nanobots, as a poster instead of a framed painting) and featured a monkey sat on a toilet eating a banana. Rimmer kept this new poster above the toilet in their cell.
When trying to set up Tim, Lister and Rimmer trashed his quarters, this monkey poster was one of the things that the put in there to ruin his date with Kochanski, along with some other of Rimmer's stuff, such as Morris Dancer Monthly. However, it turned out that it was really Warden Ackerman's quarters, and his date was actually Patricia Carling. ("Krytie TV", Series VIII)
The latter poster was included in the Series VIII DVD animated menus.