^Not if Kryptonite works the way it's supposed to. My understanding is that Superman is a solar battery. He stores power in the form of electromagnetic radiation from our yellow sun and that fuels his powers. The more energy from the sun he absorbs the more powerful he gets (at the risk of burning himself out if he ever flew too close). The radiation from Kryptonite has the effect of interfering with that process and painfully forcing that energy out of him. Since under a red sun he has no energy stored, it would have nothing to force out. Since Kryptonite apparently comes from Krypton and they managed to live with it on their home planet, it probably doesn't have any effect. I do wonder, though, if it has the same effect on ordinary Kryptonians living under a red sun if they are exposed to it for a long period of time as it did on Lex Luthor, where his Kryptonite ring gave him cancer.
I don't know if it was ever detailed (though I'm sure it has been somewhere), but what would happen to Superman were he near a blue sun or a white sun?
Kryptonite is a radioactive substance. It doesn't force solar energy out of Superman's body it gives him radiation poisoning. Long enough exposure would eventually kill him. If anything exposure under a red sun would be worse since Superman wouldn't have the extra stamina afforded by his powers to fight off the K-poisoning. While the specific level of radiation makes Green K especially dangerous to Kryptonians, Kryptonite is also dangerous for humans, although they require a much, much longer exposure time. As noted abouve, Lex Luthor got cancer from wearing a kryptonite ring to ward off Superman. Kryptonite was formed as a by-product of the destruction of Krypton and THE MAN OF STEEL, some Kryptonians were dying of Green K poisoning as the substance was forming in the dying planets core.
Hermiod and Mike Farley gave a very good explanation of the Reboot Superman's weakness to kryptonite, so I'll take a crack at the pre-Crisis Superman's weakness (which I think may be back in continuity): Due to Earth's lighter gravity, Superman's incredibly dense molecular structure is what makes him vulnerable to the specific wavelength of radiation emitted by kryptonite (and what makes it harmless to humans). One of my favorite old Brave & the Bold issues deals with Metallo (who is powered by K-radiation) readjusting to wavelength to make it deadly to normal humans while he tangles with the Batman and Lois Lane.
I've just been wondering about that one myself. The idea is that Superman's cells work as solar collectors and can get more energy from a hotter yellow (F or G) star than a cooler red (K or M) star. But white (A) and blue (O or B) stars are hotter still, so under the simplest interpretation, Superman would be even more powerful. However, Elliot S! Maggin made a different assumption in his classic novel Superman: Last Son of Krypton. From Ch. 19: http://superman.nu/thebook/lsok/?chapter=19 I'm not sure whether Maggin's assertion matched what had been established in Silver/Bronze Age comics, but it stands to reason that it would have. I also don't think the book explained why it was the case. It could be that Kryptonian cells aren't calibrated to those particular wavelengths, or that the higher-energy input overloads them somehow. Kind of like putting overly rich fuel into an engine not built for it.
Lately Supes has been losing power if you shoot him with a red-solar raygun. I saw that it one of the comics, and it was in the Batman-Superman episode of "The Batman". Supes' little son, (who is Zod's actual son if I remember correctly?) wears a red-solar watch to eliminate his powers so he can be a "normal" boy. So I would imagine there is an actual effect on his cells depending on the wavelength, more than just a weak or strong amount of energy.
Superman losing power under red-sun light is a longstanding trope. In Superman: The Animated Series, it generally wasn't an instantaneous effect, but his yellow-sun "charge" gradually wore out under red-sun light. In other incarnations, I think it was more instantaneous.
AFAIK, under a white sun, Kal-El's powers would go into overdrive and probably kill him with the overload.
I did a bit of Googling... apparently Action Comics #855 introduced a new Bizarro World that was under a blue sun, and Superman had his powers there. Apparently blue sunlight does give Bizarro a "Bizarro vision" power that Bizarrifies the things he looks at, which is how he created that world. And apparently it gave Superman the equivalent ability to temporarily induce Kryptonian powers in humans, as well as boosting his own powers. (By the way, this thread is currently the 7th hit on "Superman 'blue sun'.")
Also I believe living under a blue sun would cause Superman to develop a hankering for Fruity Oaty Bars.
You all got it wrong... Under a red sun Supes weakness to kryptonite is greatly reduced. The weaker Superman is the less Kryptonite hurts him. This is addressed on an episode of Justice League Unlimited called Chaos at the Earth's Core dealing with Metallo and Supergirl. How Krptonite hurts Superman has never been revealed. I don't think it blocks yellow sun radiation becasue a red sun would have simialr effects to kryptionite. Kryptonie was formed when Krypton exploded so it was never present on the planet. I think it affects supes like uranium would effect a human. O yeah Superman has flown through the sun to supercharge his powers, so him getting burned is a not issue.....
Post-Crisis, kryptonite affects a Kryptonian power or no power. In fact Kryptonite had been building up within Post-C Krypton for centuries and was killing a lot of Kryptonians before the Planet blew up. It was his research into the radioactive mineral within Krypton that led Jor-El to discover the planet would explode soon (because the Kryptonite build-up was messing with the core of the planet).
Superman's death at the hands of Doomsday, for instance, was caused by Superman draining all of his stored solar energy. When placed in the Fortress of Solitude, Superman's battery slowly recharged, until he had enough spark to wake him. But even then, he was incredibly weak and needed a Kryptonian battlesuit. As we saw in DC One Million, Superman won't "[burn] himself out" by flying too close to Earth's sun. Indeed, Superman resided inside the sun for 8,000 centuries and emerged with immense powers on the scale of the Silver Age Superman's.
Couldn't Superman just throw a big cellphane "S" at the red sun and be done with it? I'm sorry, I coudln't resist. Is Superman as ridiculous in the comic books as he was in those movies? -Dan
Of course not. Maybe in the '50s, when comics were at their goofiest, he could approach that level of silliness, but never at that level of creative ineptitude. And for decades, comic books have been far closer to the level of The Dark Knight and the Spider-Man movies than to something like Superman IV or Batman Forever.