Game Boy: What is the purpose of instructions that don't modify anything (e.g. AND A)?

As Jeffrey Bosboom and Hans Passant pointed out on their comments, the reason is simplicity. More specifically hardware simplicity.

LD r,r' instructions copy the content of source register (r') to destination register (r). LD r,r' opcodes follow this form:

        -------------------------------
BIT    | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
        -------------------------------
OPCODE | 0 | 1 |     r     |     r'    |
        -------------------------------

Destination and source registers can assume these values:

 -----------
| BIT | REG |
 -----------
| 111 |  A  |
 -----------
| 000 |  B  |
 -----------
| 001 |  C  |
 -----------
| 010 |  D  |
 -----------
| 011 |  E  |
 -----------
| 100 |  H  |
 -----------
| 101 |  L  |
 -----------

In order to implement these instructions in hardware we just need a multiplexer that receives bits 0-2 to select the source register and another multiplexer that receives bits 3-5 to select the destination register.

If you want to verify if bits 0-2 and bits 3-5 are pointing to the same register you would have to add more logic to the CPU. And as we all know, ressources were more limited in the 80's :P

Please note that loading instructions such as LD A,A, LD B,B, LD C,C, LD D,D, LD E,E, LD H,H, and LD L,L behave like NOP. However AND A and OR A DO NOT behave like NOP, since they affect the flag register, and their execution might change the internal machine state.

Instructions like LD A,A and AND A may appear to be NOPs but they might also change the processor flags and be used for testing the value of a register.

Be sure to check the instruction set documentation carefully for such side effects.

There is actually purpose in AND A (as well as OR A) instruction -- it sets flag Z when A is zero and clears otherwise. So both AND A and OR A are frequently used for this purpose.