Walt was later attacked by a polar bear in the jungle. He hid inside a banyan tree while the polar bear rammed at him. During this encounter, one can allegedly hear the polar bear say "Walter" as it swipes at the tree. ("Special")
The Latin name for "polar bear", Ursus maritimus, is mentioned on the blast door map, implying that the bears were used during experiments on the Island. ("Lockdown")
STATED GOAL, REPATRIATION ACCELERATED DE-TERRITORIALIZATION OF URSUS MARITIMUS THROUGH GENE THERAPY AND EXTREME CLIMATE CHANGE
While chasing Dave, Hurley ran into Charlie and Mr. Eko and asks, "Did either of you see a guy run through here -- in a bathrobe, with a coconut?", to which Charlie replied, "No, but I did see a polar bear on roller blades with a mango." ("Dave")
Charlie told Locke that he used to get high and watch nature programs on the "Beeb" (BBC), where he learned that polar bears are very clever and "are like the Einsteins of the bear community." ("Further Instructions")
When Phil catches Jerry and Rosie dancing and flirting in the Dharma security room, and Phil says that Rosie shouldn't be there, because it's their watch, Jerry jokes about that the worst thing that can happen is the polar bears are going to figure a way out of their cages. " ("LaFleur")
When Pierre Chang finds out Hurley knows about the dead body in the back of the Dharma Bus, he threatens to assign Hurley to a job "shoveling polar bear feces" if he told anyone. ("Some Like It Hoth")
Originally, director J.J. Abrams had been vehemently opposed to using a CGI polar bear on the show. The first version had a stuffed prop bear shot out of an air cannon towards Sawyer as he pretended to shoot at it. This version that was shown on the pre-broadcast commercial before the pilot aired. Someone posted a freeze-frame capture of this online, which according to J.J., looked "laughable and ridiculous". He conceded to trying the CGI version, which he and the production team liked much better. This was the version that ended up in the broadcast airing of "Pilot, Part 1". J.J. remarked that he was surprised people would analyze the show in such detail. The two versions can be seen below for comparison.
Polar bears are excellent swimmers that have been known to swim up to 200 miles 
The polar bear may be a reference to ironic process theory. From wikipedia, a classic example is Fyodor Dostoevsky's quote from Winter notes on summer impressions: “Try to pose for yourself this task: not to think of a polar bear, and you will see that the cursed thing will come to mind every minute.” 
From the RuneScape Wiki, the wiki for all things RuneScape
A polar bear, hiding in a well.
Polar bears are polar animals that have only been seen in situations with penguins so far. Although not technically animals, the Arctic bears from the Summoning skill resemble Polar bears. However, their level 71 Summoning requirement is implemented when talking to Nanuq, who requires the usual 10 levels above the minimum to talk to.