If the character were being damaged by an attack that does fire and cold damage, and they were immune to cold damage, they would take the fire damage.
Recently errata'd rules made it clear that characters making attacks with one keyword and using a weapon with another keyword had to choose one keyword to apply to powers. I think the only way a character can achieve two keywords now is that:
a) they take the Energy Admixture (covers one power)
b) the power itself has multiple keywords
Multiple keywords are easy, actualy; they're just not germane to the topic. There's no contention about what happens if you get hit with an attack that does separate damage of different types -- the attack benefits from multiple vulnerabilities (and can't benefit from any given one once; 3 radiant + 3 radaint is 6 radiant, but 3 radiant + 3 fire and cold stays separate unless it hits a resist(all) 5, in which case 1 damage (your choice? frankly, I'm not sure [ ooh, that's a fun one. You hit an undead which has resist(all) 10, vulnerable radiant (5), and regenerate (turned off if it takes radiant damage with 3 radiant damage and 10 fire damage [one damage source]. Have they taken radiant damage or not?]) slips through, but each damage chunk is separately resisted/checked on immunity.
That said, it's not that hard for damage chunks to have multiple types. Admixture does it, lots of power does it, and hell, plenty of features do it (Radiant One, perhaps? Radiant and Necrotic on everything you have CA against!). I'm not really sure what RAI is (and would slightly lean towards "the damage should slip through"), but it's not a big deal, and if Wizards has intended to have immunities not zero out damage with multiple types, they haven't done that yet. "it's also fire damage" isn't an other effect -- it's the -same- effect. The one you're immune to.