Which i7 processor supports up to 1600 mhz of ram

all i7's use the same ram.

you have to check if the motherboard supports this frequency.

RAM compatibilty doesn't matter so much on the CPU but on the motherboard. All the I7's will work with pretty much all RAM.

sandy bridge platform:
you'll need a chipset that supports ram overclocking e.g. for sandy bridge cpus, p67 and z68 chipset motherboards. with the motherboard, you'll have to use intel extreme memory profile or xmp to use ddr3 1600 ram. if you use any cpu with chipsets that don't support ram overclocking, it will use default speed - for sandy bridge it is ddr3 1333 i.e. if you use core i7 with h67 motherboard with ddr3 1600 ram, the ram will run at ddr3 1333 specs.
btw if you use a core i5 2500k (cheaper than the core i7) or a core i7 2600k, you will be able to overclock the cpu as well as use faster ram with a z68 motherboard.

i'd also like to point out that there's is not much real world difference between 1333 and 1600 so you could happily use it at 1333 and not notice anything, in fact you may be able to run it at better timings.