Whatever resistances and immunities you decide it should have
This is a topic of disagreement inside Paizo's development team (and the community). There are those who are in favor of stripping the subtype immunities, and others who are against and defend that a creature should keep immunities from their subtype, and we have seen published creatures that attempt defend both sides, but none actually does.
One side defends that if a creature becomes a skeleton, it also loses the defensive abilities gained by the subtype (Immune to Fire damage for the fire subtype), while still being a creature of that subtype whenever it matters (like spells). For both subtypes at topic here (fire and cold), this interpretation makes little sense as that is all that the subtypes do (grant immunities and vulnerabilities), but for other subtypes it makes sense, like a white dragon becoming a burning skeleton, its a creature made out of flamming bones, why should it be immune to cold? The template immunities and vulnerabilities should take precedence over the (now lost) subtype.
While the other defends that the immunity gained by the subtype is not a defensive ability and is not lost, or that the type (and subtype) takes precedence over the vulnerabilities of the template. I personally disagree with the reason behind this logic, but it's the one that makes most sense when you consider the example from before (burning skeleton white dragon).
There is one published adventure that has an example of immunities stacking, on the City of Golden Death Module, where we are presented with Tar-Baphon's Dragon (SRD reference without the flavor). He was a Young Gold Dragon (Immune to Fire, Vulnerable to Cold) that was killed and returned as a Skeleton (Immune to Cold). The final creature is immune both to cold and fire damage and is the dragon on the cover of the module.
However, he is labeled as an "unique dragon" (as opposed to being listed as a dragon with the skeleton template) since it didn't become a mindless skeleton and retained many of it's special abilities, including spell-like abilities and the breath weapon. As such, not a completely valid example for this, but this is the closest thing published by Paizo.
There are cases of creatures that lost an immunity once they got a template, like the Zombified Storm Giant (SRD reference) from Carrion Crown #3: Broken Moon. But here, the immunities are granted by the creature, and not her subtype, which is exactly what the template says and thus it was lost.
But why would an immunity granted from a type, or subtype, which are nothing more than design shortcuts to a list of abilities that are common to different creatures (and unnecessary copy+pasting), be different from an immunity granted by the creature? In my experience, when they have to publish a creature that would cause problems like this, they make up a new creature and call it a day. Rarely we see creatures with templates that will simply point out to a bestiary book, and when they do, those are simple templates (like the advanced template) that shouldn't take much preparation from the GM to apply on the base creature. From a designer perspective, it is much better to create a new creature if you will already have to write full a statblock for them and avoid being inconsistent with the rules.
There is one published template that attempts to address both subtypes, the shadowfire creature from the Emerald Spire Superdungeon. The template specifically replaces the vulnerabilities the base creature had and add immunity to both cold and fire damage, as it becomes a creature of blackish fire. The creature gains both the fire and elemental subtype if they did not have them already (the example in the book are fire elementals).
Let's not even begin to talk about when a published book completely ignore the rules and create something new for the rule of cool, like the Lycanthrope Ghouls from Classic Horrors Revisited, which also adds both Fire and Frost Giant ghouls with the relevant subtypes and immunities from each.
Lycanthrope: While a ghoul cannot become a lycanthrope, a living lycanthrope who succumbs to ghoul fever could rise as a ghoul. In most cases, this transformation removes the lycanthropic curse, resulting in a standard ghoul, but in rare events the resulting monster is a true ghoul lycanthrope. To create stats for such a creature, simply apply the lycanthrope template to a ghoul—this is an exception to the general rule that you can normally only add the lycanthrope template to a humanoid.
In the end, you must see for yourself which is best for your creature and nothing in the rules forbids you to make a red dragon skeleton that immune to fire and cold, followed by another that is immune to cold and vulnerable to fire, and a third one that is immune to fire and vulnerable to cold. You could even create one that is immune to all energy types and still be a valid creature under the rules. You might want to adjust the CR by +1 though (seriously).
