AES-encrypted 128-bit null block: lookup table?
In practice we use something called a block mode.

The traditional block mode is cipher block chaining (see image). In this block mode we XOR the ciphertext of the a block with the plaintext of the next block. The first plaintext block will be XOR'd with a public initialization vector. This way we make sure that every ciphertext is different.




  How isn't this a weakness?


It is. In your case the block mode that is used is ECB, basically applying the block cipher to each block independently. ECB is considered insecure for normal use.


  Could you make a lookup table for all blocks of only zeros?


No. Because almost always keys are generated randomly, we would need to store $2^{128}$ entries in our lookup table. Our universe will never have enough disk space to store it.