What can cause an itchy rash on hands and feet?

ITCHY HANDS/FEET. New soap or detergent. Medications you may be taking. Athlete's foot with spread to hands. Have you taken any medications such as antibiotics lately. If it persists see your dermatologist.
Many things. There are many different causes of a rash like this. You would need to seek care with your doctor to determine what it might be from. Rashes are something that need to be seen in order to determine the best diagnosis and treatment options.
Various. Most rashes that are itchy are due to exposure to allergens. This can be worse in those with underlying eczema. As long as the skin is not broken and infection seems absent, you can try over the counter cortisone application to lessen the symptoms. If persistent, spreading or causing deeper lesions, a dermatologist should be consulted.