Causes:
- A Sprain is an injury to a ligament, the tissue that links bones together at joints. Sprains happen most often in the ankle, knee, elbow, or wrist.
- Strains are tears in muscle tissue. They happen most often in the muscles that support the calf, thigh, groin, and shoulder.

Signs and Symptoms:
Strains:

When a sprain occurs, blood vessels will leak fluid into the tissue that surrounds the joint. White blood cells responsible for inflammation migrate to the area, and blood flow increases as well.

Swelling damages cells by decreasing oxygen supply.

Compression wraps can push the swelling away from the injury which is good.

• Muscle stiffness, tenderness, or soreness
• Swelling

• Pain at the time of injury
• Swelling
• Bruising

Treatment:
All but the most minor strains and sprains should be evaluated by a health care provider.
RICE - Rest, Ice, Compression, and Elevation

• Rest the sprained or strained area.
• Ice for 20 minutes every hour. You can reduce frequency over 2-3 days.
  Cold numbs the nerves and constricts the blood vessels, reducing swelling and tissue breakdown, and easing pain.
  The time-honored cold pack for a sprain or a fresh, swollen bruise is a bag of frozen corn or peas.
  Cold reduces metabolism and allows cells to survive temporarily with the lack of oxygen.
  After 3 days you can start using heat.
• Compress by wrapping an elastic (Ace) bandage or sleeve.
  It pushes the swelling away from the injury.
• Elevate the area above heart level.
• Give an over-the-counter NSAIDs (non-steroidal anti-inflammatory drug) like aspirin, ibuprofen (Advil®, Motrin®), and naproxen (Naprosyn®, Aleve®) or Acetaminophen (Tylanol®).
  However NSAIDs may increase bleeding by decreasing platelet function.

Alternating ice and heat (Contrast therapy)
Contrast therapy, which consists of alternating applications of heat and cold, is also used to treat athletic injuries, although the benefits have not been proved.
A cold-to-warm ratio of 1 to 3 min or 1 to 4 min appears reasonable based upon clinical observations and experience.
Many have suggested that contrast therapy results in cycles of vasodilation and vasoconstriction, thus creating a pumping action to reduce swelling. However, tissue temperatures are not affected by contrast treatments (Myrer et al. 1994, 1997; Higgins and Kaminski 1998). The brief exposure to cold and the fact that superficial heating has minimal effect on deep blood flow suggest that there is little vascular response to contrast therapy.

Even though there is no good explanation for the effects of contrast therapy, this approach has been used to treat some physically active individuals. For example, contrast therapy may be effective in reducing edema (swelling) in subacute foot and ankle injuries.

As with many therapies, there has been little investigation of the effectiveness of contrast treatments. Cote et al. (1988) reported that swelling increased in sprained ankles after contrast therapy administered over 3 days after acute lateral ankle sprain.

See the article for more. But my take was if swelling and/or pain gets worse after heat, then don't use it.
Source: How to use heat and cold to treat athletic injuries at Human Kinetics