Whether your eyebrows are tattooed or “microbladed” (which means that your eyebrows are still tattooed), either way you’ve made an investment that you want to protect by prolonging its longevity. If you have ever seen body art tattoos on an older person who had clearly gotten those tattoos when they were much younger, you will realize that even though they are indeed permanent, they are nevertheless subject to fading. One key factor that causes fading of course is UV light exposure, and so one way to protect against premature fading then is to use sunscreen. While this is true and will help with some of the fading, it will not stem the inevitable fading due to the continual shedding and regrowth of the skin and hair.
Keep in mind that there may also be a cosmetic tradeoff to wearing sunscreen over your eyebrow tattoos, particularly if they are darker-colored. Similar to some foundations or face powder, some sunscreens may cause color to take on slightly different hue, usually looking more ashy. If your complexion is ruddy to begin with then that may not be an issue, otherwise the color of your eyebrows may not look as natural sitting underneath a layer sunscreen or makeup. Much like people who prefer to use slip covers over their chairs and sofas to protect the original material underneath, the trade off is in deciding how you prefer to enjoy your eyebrows.