TL;DR It's a superpower; you decide!
Poisons are a general term for a range of substances that are harmful to living creatures. Every poison can attack the body in different ways; but I think, for the sake of your plot, you should differentiate between things that harm through specific chemical pathways, disrupting the body's processes (i.e. a classic poison) vs. things that are just caustic and destructive, like lye or acid. The resistance to poison really depends on how you set up the mechanism; since poison is just a general term for things that harm in millions of different ways, it's a bit difficult to create a scientific method of eliminating them, although you can definitely come up with one. You can always go with the 'traditional' magic route, and it's up to you to decide whether it's some magic that is infused everywhere and hangs around the cells, or only applies when the cells remain attached to the rest of the body (i.e. not a tissue sample). I can suggest though, two "scientific" mechanisms for this to take place. The first is that in a person with this superpower, every cell has an organelle responsible for defense against poison. It simply monitors the cell to check if it's behaving normally. If it's not, then it can detect uncommon chemicals that don't belong in/around the cell (likely poison) and signal the immune system to quickly create the appropriate antigens to disable/eliminate the poison. The people with this superpower will also need an extremely fast immune system to create antigens in time (this can normally take days). In this case, this will not work for tissue samples, because there is no immune system to create antigens. It can detect them, but can't fight them. I think this makes a bit more sense than the next option, because it relies on the existing framework of the immune system to fight the poison. However, if you want it to work in samples, then you can perhaps say that the organelle responsible for defense against poison can also analyze the poison and release RNA with instructions for the ribosomes to create antigens. A lot harder to pull off, because normal cells don't do this; the immune system does. However, this could work even in tissue samples.
So there you are. I've said 3 options for how this superpower could work. You could go with magic, or 2 of the scientific methods I've suggested. Of course there are many more mechanisms that can be thought of, you can come up with your own if you don't like mine. But the answer to your main question of whether or not this will work in tissue samples, this depends completely on the process. It's your decision of whether or not tissue sampling will work; based on that. choose a process that fits your storyline.