There are several ways to remove a tight ring off a swollen finger short of cutting the ring. The simplest way is using liquid soap, or any available lubricant, such as hand lotion or vaseline. Once applied, the ring can be rotated around the finger in circles while, very gently, advancing it towards the finger tip. It is very important to prevent the skin on top from wrinkling as it will block the advancement. To do this, it is best to bend the finger at the knuckle just in front of the ring. Other tricks that can help with this technique are to elevate the finger / hand above the head for 10-15 minutes before it is done, or immersing it is ice water for a few minutes.
It is very important to remove all rings after any type of wrist of hand injury before swelling and increased pain makes it hard to do so.
If the above does not work, this video shows a neat trick.