Age spots are not cancerous.
More Info: Lentigines, better known as liver or age spots, are not cancerous nor do they lead to cancer. That doesn't mean that they should be ignored. It's not the age spot that's harmful, it's the misdiagnosis of the age spot that is. 
What Are Age Spots?
The term age spots is actually a misnomer as anybody of any age can get them. You can be predisposed to them through heredity and you can also get them as you age through sun exposure. Age spots are a buildup of melanin, which is the skin's response to sun exposure. Melanin turns the skin brown to protect against sun damage but can begin to clump as you age resulting in a dark spot on the skin. 
The Dangers of Self-Diagnosis
The reason that you should have age spots checked by your doctor even though they are not cancerous nor do they lead to cancer is that they may be accompanied by actinic keratosis, which are precancerous, red, scaly elevations of the skin.  Another compelling reason to have your age spots checked out is that skin cancer can be mistaken for an age spot by the untrained eye. Accounting for 10% of melanoma cases, lentigo maligna melanoma often occurs on sun-damaged skin, and is frequently mistaken for an age-spot. Easily treatable in its early stages, this type of melanoma is dangerous as it often goes undetected. Even superficial spreading melanoma looks like a flat spot resembling a freckle in its early stages. 
Early Detection=Higher Cure Rate
Melanoma accounts for 77% of all deaths from skin cancer. With early detection, 95% of melanoma cases can be successfully treated.  You should perform regular examinations of your skin and consult a physician anytime you find something suspicious. According to the American Academy of Dermatology, there are a few key signs to watch for when examining your moles and areas of pigmentation that include asymmetry, where one side looks different from the other, an irregular or poorly defined border, has variations in color, or are larger than a pencil eraser. 
Resources for Early Detection of Melanoma
Though you should never use the internet to self-diagnose any condition, there are some wonderful resources to better educate yourself about melanoma. Here are just a few.
If you have never seen melanoma, Mayo Clinic provides a slideshow to help identify skin cancer.
The Melanoma Education Foundation provides a tutorial with step-by-step instructions on performing a melanoma self- examination.
The ABCDEs of Melanoma, created by the American Academy of Dermatology, is an easy way to remember the skin changes that you should be watching for.