Can rainbow tables be used to find private keys from public keys? I understand that the public key is kept hidden until a bitcoin transaction is made, at which point the public key has to be revealed to prove that you created the transaction with your private key. Until then, only a hash of your public key is ever shared, know as your bitcoin address. 

Once my public key is known, would it be possible for someone to lookup my private key in a rainbow table ?
No.

A rainbow table is simply a memory-cpu tradeoff that allows you to find preimages of a function faster by having a precomputed table with some of the entries.

In order to construct a rainbow table for bitcoin private/public keys, you still have to first iterate through all the keys at least once. It doesn't matter that only part of the keypairs will end up in your rainbow table - you still have to go through all them.

That is an impossible task. There are 115,792,089,237,316,195,423,570,985,008,687,907,852,837,564,279,074,904,382,605,163,141,518,161,494,336 valid private keys to go through. To put that in perspective: if you would put a computer on every square micrometer of earth (1/1600th of the cross section of a human hair) that can compute as many public keys per second as the entire Bitcoin mining network does hashes per second at the time of writing (4 exahash/s), it would take you 120000 times the age of the universe to go through all keys.

And, even if somehow you had an oracle that could give you the rainbow table, it would need to be impossibly large to make cracking private keys practical. If your table had a trillion entries, it would only make cracking a trillion times faster. That's still utterly impossible.