If the only problem is the keyboard, Apple will replace it for $180. 1% of the time.
99% of the time, they'll say they see liquid and charge you $750-$900.
They do this to subsidize all the people who get free stuff because they lie about spill damage to their computers. By lying to them, you are continuing to foster an industry for independent repair technicians who make clients out of people told $700-$900 by Apple. So, from my personally owned business point of view, sure, lie to them
As a moral human being, don't lie. They're going to see it. Corrosion is so obvious, and you're going to look dumb if you tell them you never got water inside of it when they see big green blobs of **** in there. Do you really think the dude in customer service hasn't had 900000000 people before you lie about it? He's probably at the end of his rope, and the second you start lying they're going to lose all respect for you and treat you like crap.
Just tell the truth, and have comfort in the fact that all around the country there are people just waiting for you to leave Apple with a $700 quote so they can be the ones to fix your machine for $100-$200.
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I got a MacBook Air working again by cleaning all the corrosion off with 99% alcohol and a kids' toothbrush I bought at the dollar store. I'd also pick up a magnifying glass to really look for trouble spots.
The keyboard had shorted and a new topcase got the computer booting again, but no battery. Still working on that one. A replacement magsafe board didn't fix that issue, so maybe a new logic board.
iFixit has a good guide for taking it apart.
Did you try a new battery?