Some teams do insure certain contracts, however many contracts are too risky for any insurance writer to issue a policy (A great example is Amare Stoudamire's contract with the Knicks).
The issue that necessitates player insurance in the first place is the idea of guaranteed contracts. These are contracts that require the team to pay out the entire value (or some major percentage) of the contract no matter the status of the player. If the player is hurt or is cut by the team they must pay all of the guaranteed money to the player.
However different leagues have different types of contracts and different levels of guarantees based on (At least in the US) how their collective bargaining agreements are structured. These are the agreements that allow the leagues to exist and play games in the US. In the four major sports here are how contracts work:
MLB: Most every free agent contract is guaranteed. Initial contracts are not guaranteed (unless an extension is done) nor are arbitration settlements.
NBA: Contracts are mostly guaranteed, although not all are and some contracts are only partially guaranteed.
NHL: Not entirely sure about rookie contracts, but free agent contracts are for the most part guaranteed.
NFL: Only the signing bonus and first couple of years are guaranteed on NFL contracts (and depending on the contract, those first years may not even be). If a player is cut during the off season they do not have to pay the remainder of the contract (once a certain date hits the player's contract is guaranteed for that league year).
Now to the question of insurance. Premiums for sports players can be very high, as much as "10% of the annual value of the contract." But may be worth it in the case of very high value players. They also typically do not cover areas of previous injury.
So suffice to say that yes, major sports teams do take out insurance for high value players, but those policies are very expensive and can be of limited use if a player aggravates an old injury. And players with extensive injury histories are basically uncoverable.