Ink Color Limitations
Regarding inks, there is a huge variety of colors already made, and you can also potentially mix inks too (pure pigment inks are alcohol-based and cannot be mixed, water-based inks can be mixed. Mixing is generally not recommended as it's a complex process and can give terrible results if not done well).
The important thing in my opinion here, is how you choose the correct colors according to the skin tone. Bolder colors like reds, pinks and purples look usually good on fair skin, while yellows and orange can be a bit more tricky. Bright blues, oranges, yellows, greens, and purples look better on medium or tan skin, and very light tattoo ink colors are the best choice for darker skin (light pinks and blues). In this case, lighter than the skin or much darker than the skin are usually the best ways to go.
Line Widths and Fine Details
This is probably the main thing you will have to keep in mind when designing a tattoo. The basic premise would be: The more detail you want, the bigger the tattoo will have to be. There are very thin needles that can work with very thin lines, but there is an 'extra' problem you have to consider: Lines get thicker with time (up to 30% thicker after a year or two). So even if they look great when the tat is new, this won't always be the case.
Body Shape, Uneven Surface of Skin and Context!
This one is probably more difficult. First question would be: Is the surface tattoo-able? Burnt skin, for example, can be very sensitive and difficult to work in. It will depend on how old the scar is, but in general scarred tissue doesn't hold ink as well since it's more dense than regular skin.
Something that is quite essential when designing a piece is where it will be located. This is, in my opinion, the most important consideration apart from colors and detail level. Art is 2D, bodies are 3D. The same design can look very different depending on where it is: Back, upper arms, legs, ankles. Even if the 'shape' is right, you also have to consider if the person has other tattoos, and how the new one will interact with them.
(Note: I'm not a tattoo artist, but I'm a tattoo enthusiast and have a few of them)