I've had 2 MBA's with the startup problem (3 beeps). After reading all the answers above there are different solutions. Common problem is the memory but it is not always a hardware problem like disturbed connections, etc. In my experience (both MBA's) it had to do with a software problem: Failure with the sleepmode. It's the swapfile which fails in a kind of way. If that fails the MBA won't boot and reacts with 3 beeps. I've tried to startup in all thinkable ways, mentioned above. Sometimes they booted up. If it was a hardware problem they should never boot so it must be something else.
I discovered that when you can make the MBA complete out of power in all components (especially the SMC) it will boot normally. Taking out the logic board or recharge the thermal paste or baking in the oven, all methods make the logic board complete out of power. That cleans the swapfile and takes away the startup problem. After reassembling the MBA's did boot normally without the 3 beeps.
It's my experience and not THE solution but just an opportunity. Good luck and please respond if you have any comment, positive or negative. We can all learn.