You have three layers of skin, the epidermis, dermis and hypodermis. The ink of a tattoo sits in the middle dermis layer once healed. This layer is below the epidermis, which is the bit that ‘sheds’, which is why your tattoo never wears off. The skin on your palms and soles have a lot more of these cellular layers which means faster shedding of the epidermis layer, much quicker than anywhere else on the rest of your body.
Elsewhere on your body the epidermis layer is of a fairly consistent depth, making it possible for the experienced tattoo artist to accurately judge the depth the ink needs to go to in order to hit the dermis layer. This is not the case however with the palms of the hands and the soles of the feet. The epidermis layer is thicker here than elsewhere, and also varies hugely in depth across its surface. The result is that it is almost impossible to get the ink into the dermis layer reliably. The ink that doesn’t go deep enough will eventually ‘shed’ from the epidermis layer (often mistaken for the tattoo wearing off), which will mean that your tattoo is likely to require frequent retouching every few months. However if the ink goes too deep, through to the hypodermis, it causes what is known as ‘blow-out’, where the ink spreads through the capillaries under the skin in a blotting paper type effect.
Tattoos are going to be more painful in less padded areas of your body, which is often where the skin is more sensitive. The palms of your hands and soles of your feet have almost no padding, and are one of the more sensitive areas on your body. Due to the amount of nerves you also have in your feet, if you find a tattooist who is prepared to tattoo the soles of your feet, keeping still throughout the process is going to be particularly difficult as it is one of the most painful places to be tattooed.
Healing a sole of foot tattoo is also problematic; as your tattoo goes through the healing process it will develop a scab, which is a protective layer made up of blood exposed to air, formed by platelets. If any of the scab is rubbed or picked off before the skin underneath has fully healed you stand a good chance of not only loosing an area of colour but also of any line work. Ideally you would need to be bare foot throughout the whole healing process. Which in itself raises further healing issues with respect of keeping the area clean. You need also to be prepared for localised swelling too, which you will need to take into account and be prepared for.
While healing palms of hands is slightly easier, you would have to avoid soaking your hands and also ensure that they are kept clean. Because of day-to-day activities hands are more likely to pick up unwanted bacteria, which could potentially lead to infection.
In conclusion this is why it is next to impossible to get a good tattoo on the palms of your hands or soles of your feet, which is why most studios will avoid it altogether.