Many deer survive quite well on three legs. After being hit by a car, a limping deer will have adrenaline running through its body, which will make it flee and disappear and you won't find it; or it will get up, look for somewhere nearby to hide and it will collapse without wanting to stand up. Even if your vehicle isn't irreparably damaged, hitting a deer can cause headlights to break, windshield damage, crush the bumper, and more. Make sure you have the right amount of car insurance coverage in case costly repairs are needed.
Keep in mind that this is only true if you only hit a deer. If it veers and damages another vehicle or property, then yes. You'll need to contact your insurance company, local police, etc. as you would with any other accident.
If this isn't the case, it's entirely up to the driver to decide, but most states ask him to notify the authorities if the deer is seriously injured. It can be a potential hazard to anyone nearby and should be handled as soon as possible. If you hit a deer with your car, it will be considered a comprehensive claim. Any comprehensive claim will be on your record, but generally won't affect your premiums, since you are not at fault.
If you receive a high frequency of no-fault claims, your insurer may require you to increase your deductible to cover the cost of the damage. Generally speaking, fault claims can increase your insurance premiums more adversely, unless your accident is condoned. If a collision with the animal seems inevitable, the safest option is to hit it while maintaining full control of your car. The alternative of veering into oncoming traffic or leaving the road surface could be much worse.
Motorcyclists are especially at risk when they collide directly with animals.