An aircraft would be able to land without an engine switched off, however it would a) be illegal, and more importantly b) be very unsafe. Think about it this way. You're on final, about to land, and another aircraft enters the runway without clearance. When this happens, you have to go-around. Now, if you have one engine shut down, you are going to have a lot of trouble doing this.
A twin should be able to climb out with one engine inoperative, but you wouldn't want to be doing it by choice - your climb gradient isn't going to be flash and you are going to have to throw in a lot of rudder to counter the yaw from the assymetric thrust. If you are below a certain speed (Blue Line it is referred to in light twins, I guess it would be the same in a larger aircraft), the aircraft may not be controllable at all.
Even worse, if you were to lose the engine which was running for some reason, then you would be in quite a predicament, especially if you needed to go around. Jet engines can't just be switched on and develop full power instantly - they take time to spool up.
(Unrelated to the question, but the lag in engine response is also a reason why they run the engines up partially just before going to full power on a take off - because of the lag in response, it may not become evident that an engine isn't producing power until the other engines are all at full throttle, and by then it will take a while to bring them back down. Again, not a nice situation to be in. Aircraft have been lost because of this in the past.)
Back to the question, once they are on the ground (and I would expect clear of the runway), it is a different story. I am aware of at least one airline which taxis ATR 72s back to the terminal on a single engine. But you should never see it being done in the air. It is simply too reckless.
"So powerful is the light of unity that it can illuminate the whole earth." - Bahá'u'lláh