Or is that not a sun at all as it is a giant ball of dry ice. or is it a diamond struture generating immense heat through friction? Help me out here sci, please.
If you had enough ice, my thought is that it could come a star. But it would need a lot of energy to start. I'm not sure if gravitational compacting of ice (or really, it would become water far before it became a star).
Frictional energies are not enough to create star-like heat output. Burning organic material like wood or gasoline could potentially produce star like heat, but it would only be able to do so for several hundred to several thousand years at the size our sun is (and you'd need to supply oxygen, as space doesn't have enough floating around for things to burn).
If you're interested in how stars work, try to visit the wikipedia page.
I actually doubt you'd be able to do it with ice. Gravitational compacting will only get you so far; you need the thing to be really dense in order to get a sustained nuclear reaction.
You might get some fusion in the core, but doing so will vaporize the ice around the fusion region, trapping vapor inside a shell of boiling water inside a shell of ice; this in turn will lead to fissures in the ice as the vapor and water escapes, Eventually as material continues to escape the core will depressurize and cool and you'll be left with a big shell of ice.
a star is just a ball of hydrogen/helium fusing together. out of all the elements, helium requires the most heat to change between states of matter. so no, you could not have a frozen star. you could have a block of ice the size of a star though with planets in it's gravitational well.