From what I can remember (I'm at work now, I'll check properly on my 88CM when I get home), the Kiev bodies have no DOF lever, but the lenses do. All metering and focussing is done wide-open, and then it should close the aperture when you release the shutter. It's the camera that opens the aperture, and a spring in the lens closes it when the camera lets go.

Firstly, if the aperture is not closing when you release the shutter, you have a problem (although you probably already know that).

If you press the DOF lever on the lens, and the aperture closes, does it stay closed when you release or spring back open? If you press the shutter, does the aperture stay completely wide-open, or does it get very slowly narrower?

Take the lens off and just work the body, you should see the little pin inside the left (looking at the mount) jump back when you release the shutter. If not, there's a body-problem.

If that pin (in the body) does jump back when you release the shutter, it's probably a lens problem.

Just look at the lens and press the pin coming out of the lens at different aperture settings. If pressing then releasing the pin doesn't make the aperture close (or doesn't close quickly enough), you have a problem.
I've taken apart my fair share of lenses, and more than likely the problem is that the mechanism is gunked up with old oil, or a spring is broken (or just stretched past the point of it being a spring anymore, although that should only happen if you bork it yourself taking it apart, ask me how I know).

If the lens is borked, then there's 4 solutions. Fleabay a new one. Get a CLA from someone reputable (and pay more than fleabaying a new one). Take the lens apart and try to de-gunk it. Get a Pentacon Six where you can move the camera-body-pin out of the way so it's always in stop-down metering mode, then use the DOF-lever on the lens to close the aperture and it'll stay there.
If it's the body, I've never pulled one apart (yet), so I don't know how hard it'd be to fix.