Galois Field ($2^{16}$) for encryption in place of $GF(2^8)$ in AES AES algorithm is using $GF(2^8)$ Galois Field. What if we use $GF(2^{16})$ in place of $GF(2^8)$? Will this improve the algorithm or not?
The result would not be AES, but Rijndael  a generalization of Rijndael. The Rijndael originally was proposed with 1 block size (128 bits) and 5 key sizes (from 128 to 256 is jumps of 32 bits). In the standardization to become AES, only 3 were selected (jumps of 64 bits).

What makes block size to be 128? Each of the internal operations does its contribution. The wordsize is 8 bits (That's why the Galois Field has this size), and there are 4 columns and 4 rows. That makes $4*4*8 = 128$. The key sizes difference comes from the number of columns. Each extra column, as it has 4 rows, adds 32 bits. The AES adds 2 columns for each increment.

It is possible to play with this parameters, but you'll have to study individually to each of the Rijndael's operations what this change mean. Imho, the Pseudo Random generator (PRG) of the key expansion, I think will still be, at least, as valid as it is. The subBytes operation would experiment a size explosion that may affect the performance. Perhaps shiftRows and mixColumns would be the less affected. And don't forget how many rounds would be a good number.

As a summary, would this improve the algorithm? It can, if it comes with the appropriate adjustment of the other parameters. But then, if you implement it, you'll be away of the standard.