While Bender is a notoriously prolific drinker, we learn that this habit is actually how Bender fuels himself, and without drinking alcohol he will grow a rust beard and act like a human would when drunk.
Do all robots in the Futurama future require alcohol to operate (and avoid rust)?
No, there are a number of other robots that don't appear to run on alcohol:
Robot 1-X: This is the newer model robot that runs on pollution and emits pure oxygen with a fresh pine scent. They first appeared in the fifth broadcast season episode "Obsoletely Fabulous".
Planet Express Ship: It's a machine with artificial intelligence, so I think it counts as a robot. It was powered by dark matter for much of the series, but now that dark matter is useless it runs on whale oil.
Project Satan a.k.a. the original Were-Car: This car/robot is powered by Ed Begley, Jr.'s electric motor, "the most evil propulsion system ever conceived". (episode: "The Honking")
Lisa: She's powered by a water wheel. Although she is first seen in Bender's imagination, she shows up in other episodes, proving these robots actually exist.
In addition, it is possible for robots to run on mineral oil instead of liquor, as Bender did in the episode "Hell Is Other Robots" since drinking alcohol is considered a sin by followers of Robotology. In the episode "Mother's Day", Greeting Card advocates banning alcohol, calling it "the opiate of the human bourgeoisie", which would further suggest that there are alternative viable fuel sources, like mineral oil.
All robots in Futurama do not require alcohol to operate. But some of Mom's Friendly Robot Company do seem to require alcohol to function properly.
Bender operates on alcohol-based fuel, which is converted into electrical energy strong enough to power not only Bender himself, but also small household appliances plugged into his power receptacle. Having low alcohol level in alcohol-fueled robots made by Mom's Friendly Robot Company impedes production of electricity needed for them to function. So when Bender stops drinking and sobers up, in human terms, he becomes increasingly drunk-like and dysfunctional, and grows a red beard of rust. --Wikipedia > Bender > Sources of Energy
Bender also has a micro-nuclear reactor which we see in Godfellas when he hosts an entire civilization on his body that ultimately engages in nuclear war, exterminating themselves. Bender is the source of the radioactive material used in that war.
