Wolverine has recovered from every conceivable injury from impalement to immolation since his healing factor makes him functionally immortal. But there is one injury he is prone to and unable to rapidly heal from, a sunburn.
As the most famous mutant, Wolverine has starred in more Marvel titles than almost any other character. His gruff demeanor and penchant for violence make him a staple antihero and his healing factor grants him a longevity that puts the Canuck at around 130 years old. He has faced all sorts of foes and bounced back from most life-threatening injuries. But without sunscreen, he’d still be red as a strawberry.
A sunburn is one “injury” the Wolverine cannot beat. Sunburns are caused by overexposure to the sun’s UV radiation that burns through layers of skin cells and end up causing a rosy burn to those not blessed with an abundance of melanin. Melanin is the body’s defense against UV radiation and is a genetic trait passed down, allowing some people to tan while others with a lack of melanin burn.
Wolverine was born and raised in Northern Alberta, Canada, and is about as pasty as they come. Like many of his fellow Canadians, Logan has a distant relationship with the sun. Growing up in the wild, cold and tundra-like Alberta he didn’t have to worry about applying sunscreen to the only exposed skin not covered by a coat. His milky complexion means he doesn’t have a surplus of melanin to protect against sunburns. Though it doesn't seem to have been explored in any great detail in comics, Logan’s healing factor probably wouldn’t defend against sunburn.
Since melanin is a genetic trait, unless he was born with the ability to bronze, he likely just burns in the sun. In Uncanny X-Men #251, Wolverine is crucified in the Australian outback and is colored in a distinctly pink hue. It’s not specifically noted that he is getting a sunburn but Logan’s healing factor had been significantly worn down up to that point in the story.
Logan’s healing factor has been stretched to the absolute limit restoring him after being torched to just his adamantium skeleton and even surviving a nuclear bomb—in Wolverine Vol. 3 #43 and Venom #9 respectively. But healing a sunburn seems outside his healing factor’s capability. Since genes have more say on protecting against sunburns than any kind of mutant power, his pallid complexion is a significant weakness. With all that hair it has to be hard for the sun’s rays to even reach his skin, but Logan’s fur coat is his only protection from the sun. Wolverine won’t ever tan up and instead is destined to look like he’s blushing every summer regardless of his healing ability.