Do area specific exercises (Eg. Tummy crunches) work?

It would be nice, but sadly it doesn't work that way. As you use energy, it comes from glucose and fat in your blood.

This is very much simplified, but a starting explanation:

Here is what happens when you eat:
Net Carbs - moves into blood as glucose
Protein - breaks down to individual amino acids, which can then build new amino acids or add to muscle tissue / body protein or can also be converted to glucose
Fat - added to blood stream as fat / triglycerides

In the blood:
Glucose - either is used as energy by brain / central nervous system or skeletal muscles; stored as glycogen; or converted and added to adipose tissue a.k.a. body fat (de novo lipogenesis)
Protein - adds to muscle tissue and other body tissues (skin, hair, etc.); or converted to glucose (gluconeogenesis)
Fat - either oxidized by skeletal muscles for energy; or broken down so it can move into adipose tissue (re-esterification)

Body protein can be converted to glucose and put back into blood (gluconeogenesis) - see "In the blood" section for how the new glucose can be used or moved.

Body fat / adipose tissue can be moved back to blood - see "In the blood" section above for more details on how the fat can be used or moved. Central to your question, exercising a particular muscle uses fat or glucose from blood and it doesn't matter whether that fat or glucose came directly from food, from glycogen, from protein (gluconeogeneseis), or body fat. Your muscle doesn't care how the glucose or fat got into the blood or whether fat in the blood came from food or from any particular body fat from a particular area of your body.

For example, my arms now have the least fat but my primary exercise is running. Overall, it would appear that I've used fat from my arms to fuel my leg muscles. The fat moves into blood and is circulated to the muscles using it.