How would a CPU designed purely for functional programming be different? CPU's are to an extent designed with in mind the software that people will write for it, implicitly or explicitly. 

It seems to me that if you look at the design of instruction set architectures, they are very "imperative", in the sense that each instruction encodes an imperative style command. It also seems to me that the current instruction set architectures have evolved partly based on the type of code programmers produce.

If one would design a CPU from scratch, knowing that it would only ever run programs written in a functional programming style, how would that CPU be designed differently from existing CPU's?
Actually, it has been done: https://en.wikipedia.org/wiki/Lisp_machine

One aspect in CPU design for FP is garbage collection. GC is very important for functional languages. Common implementations require that the GC can distinguish between pointers and non-pointer data. Effectively, that means storing an extra bit along your data. This is the reason that, for example, OCaml integers are only 31 bit on 32-bit architectures and 63 bit on 64-bit architectures. Integer arithmetic then involves awkward extra shifting operations. Other languages (or other OCaml data types) may waste whole machine words for that extra bit, thus using 128 bits for 64-bit integers. A CPU that is natively designed towards GC might have a 65-bit data bus but 64-bit arithmetic.

That said, a lot of non-functional languages also have garbage collection and would thus profit from respective architectures.

Another thing that comes to mind is that memory usage of FP typically is much more scattered than that of imperative programs. Mainly because it is less natural to use arrays. In consequence, these programs profit less from caching contiguous chunks of memory. So, an FP CPU might use different caching strategies.