In "Tomb of the Cybermen", there was a device mentioned that was apparently used to revitalise the Cybermen, but did Cybermen ever need to eat or drink?
Although it's never been explicitly confirmed that Cyberman do not eat, we have never seen them eat anything in Old Who or New Who, and there's significant circumstantial evidence that they do not need to eat.
My circumstantial evidence is as follows:
1) Even in the story about a half-human half-cyberwoman, there was no mention of food.
TANIZAKI: Some elements have been augmented. Some are still human. Sensory capacity, for instance. Her breathing and hearing appears completely cybernetic. And yet...there's also bare flesh. Amazing. Perhaps 55% augmentation with 45% awaiting completion. Do you think? Or perhaps, maybe 60/40. It's fascinating.
IANTO: Can you make her human again?
TANIZAKI: You took parts from a Cyber conversion unit and modified them into a life support for her? How did you know what to do?
LISA: I told him.
Torchwood - Cyberwoman
At no point in this episode is there any implication that she eats, or that Ianto has been sneaking food down to her. Yet she's been alive for quite some time solely because of the life support system they cobbled together out of spare cyber-tech. The obligatory hint early in this episode that something's not right is an "internal power drain", not missing pieces of food.
2) It's fairly clear that Cybermen in both Old Who and New Who are almost entirely robotic, and the brain is usually the biggest if not the only organic component left.
Having no other organic material left would certainly reduce the need for food. Whether or not it eliminates the need for food depends on scientific speculation I'm not prepared to do, but it seems safe to assume that science fiction tech can preserve a human brain without any food.
KRAIL: Yes, Cybermen. We were exactly like you once but our cybernetic scientists realised that our race was getting weak.
BARCLAY: Weak? How?
KRAIL: Our life span was getting shorter, so our scientists and doctors devised spare parts for our bodies until we could be almost completely replaced.
POLLY: But that means you're not like us. You're robots!
KRAIL: Our brains are just like yours except that certain weaknesses have been removed.
BARCLAY: Weaknesses? What weaknesses?
KRAIL: You call them emotions, do you not?
POLLY: But that's terrible. You, you mean you wouldn't care about someone in pain?
KRAIL: There would be no need. We do not feel pain.
The Tenth Planet
DOCTOR: They were, until they had all their humanity taken away. That's a living brain jammed inside a cybernetic body, with a heart of steel. All emotions removed.
Rise of the Cybermen
LUMIC: The most precious thing on this Earth is the human brain, and yet we allow it to die. But now Cybus Industries has perfected a way of sustaining the brain indefinitely within a cradle of copyrighted chemicals. And the latest advances in synapse research allows cyberkinetic impulses to be bonded onto a metal exoskeleton. This is the ultimate upgrade. Our greatest step into cyberspace.
Rise of the Cybermen
You could interpret this particular line as an explicit confirmation that New Who/Pete's World Cybermen do not require any food. Unfortunately, they haven't always been perfectly consistent about the organic brains thing...
DOCTOR: Loose thinking. The trouble with Cybermen is they've got hydraulic muscles, and of course hydraulic brains to go with them.
Revenge of the Cybermen
And there's even one instance where the "organic part" dies.
AMY: And what's a Cyberman?
DOCTOR: Oh, sort of part man, part robot. The organic part must have died out years ago. Now the robot part is looking for, well, fresh meat.
AMY: What, us?
DOCTOR: It's just like being an organ donor, except you're alive and sort of screaming. I need to get round behind it. Could you draw its fire?
The Pandorica Opens
So sadly I can't base any iron-clad argument on this point, which is why I said "usually".
3) It seems fairly clear that Cybermen require power.
KAFTAN: A Cyberman would stand in that form and be, well, revitalised?
VINER: Yes, that's reasonable. These projectors were probably made to fire in neuro-electric potential. Yes, that's it, I think you're right.
Tomb of the Cybermen
I'm going to assume that "firing in neuro-electric potential" is essentially the same thing as "recharging".
DOCTOR: See? Compressed information. Tons of it. That is the history of London, 1066 to the present day. This is like a disc, a Cyberdisc. But why would the Cybermen need something so simple? They've got to be wireless. Unless, they're in the wrong century. They haven't got much power. They need plain old basic infostamps to update themselves. Are you all right?
The Next Doctor
DOCTOR: Took me a while, alot on my mind. (stands and turns around) Let's see, this ship crashed here centuries ago, no survivors, but the systems are dormant waiting for power. And then the council stick a load of new cables right on top of you. Bitey wakes up and channels the power, you start crewing up from the shop as best you can, not enough power, not enough parts.
4) In New Who, the Cybermen are sometimes described as "immortal", which would be strange if they could starve to death.
CYBERMAN: You are in pain. We can remove pain forever.
LUMIC: No, not yet! I'm not ready.
CYBERMAN: We will give you immortality.
DOCTOR: Yeah, but that's it. That's exactly the point! Oh, Lumic, you're a clever man. I'd call you a genius, except I'm in the room. But everything you've invented, you did to fight your sickness. And that's brilliant. That is so human. But once you get rid of sickness and mortality, then what's there to strive for, eh? The Cybermen won't advance. You'll just stop. You'll stay like this forever. A metal Earth with metal men and metal thoughts, lacking the one thing that makes this planet so alive. People. Ordinary, stupid, brilliant people.
The Age of Steel
From the Fifth Doctor episode Earthshock (emphasis mine):
DOCTOR: They also enhance life! When did you last have the pleasure of smelling a flower, watching a sunset, eating a well-prepared meal?
LEADER: These things are irrelevant.
DOCTOR: For some people, small, beautiful events is what life is all about!
It's arguable, and one interpretation might be that Cybermen need to eat but take no pleasure in doing so, but it seems more likely that they don't eat any more than they smell flowers or watch sunsets, that all three are useless, unnecessary, "irrelevant" activities for Cybermen.
(This Wikia page claims that "the nourishment needed by their biological parts is supplied by the life-support systems of their cybernetic bodies", but provides no evidence to support this claim.)