If as an example a = 2, then arr[a++] requires the storage of the old value of a since after evaluating a++, a would be equal to 3, while it needs to access arr[2].

I have not studied compilers, but this is what I think is the difference:

arr[i++]

1. Compiler finds i
2. Compiler finds ++ which will increment i
3. Compiler needs original value of i to access arr, so it makes a temporary clone before increment
4. Compiler increments i
5. Compiler accesses arr with value of temporary clone

Value of i cannot be used directly because it has been mutated, so a clone with previous value is required.

arr[++i]

1. Compiler finds ++ which will increment i
2. Compiler increments i
3. Compiler accesses arr with value of i

Value of i can be used directly.