First, it is important to understand some of the differences between IK and FK. When you are animating an arm in FK, you first move the shoulder, then the elbow, and then the wrists. That's more or less what we do when we move our own arms. So, all and all, FK seems a bit more similar to the way we move, with the movement starting on the shoulder. The process of animating an IK arm is very different; instead of starting the movement from the shoulder, you will work in the inverse direction. You position the wrist wherever you want it to go, and the rest of the arm kind of goes with it. Both ways of animating are perfectly fine, and, for example, when animating a character dancing, walking, or running, you can certainly achieve good results with both IK and FK.
The important thing here is to keep in mind that a realistic movement of the arms will mostly start on the shoulder. This is somewhat easier to achieve with FK, because you will will work from the shoulder down. However, it can be a little tricky with IK – the very method of animating with IK, positioning the wrist first and having the rest of the arm following it, will make it look like the hands are always leading the movement. Your job as an animator is to make it feel like the shoulder is leading, even though you are positioning the wrist first.
Another important point is that when you are animating a human body, all the body parts need to feel interconnected. When you move the spine, the arms will also move. When you raise a hand up in the air, the spine will move as well. Again, this is somewhat easier to achieve with FK. If you have the arms in FK and you move the body node the whole arm – including the wrist - will go with it, so we perceive the arms as being connected to the rest of the body. With IK arms, the movement of the spine will not automatically affect the position of the wrist. You will move the body node and the wrist will be stuck in place. Because of that, sometimes we can have the feeling that the arms (wrists) are independent from the body.
So, if you are animating with the arms in IK, you have to try your best to convey the feeling that the movement is rooted on the shoulder, and it is not the hand that is leading the movement. Basically, you have to fight the IK, and animate the wrist and the elbow just like a real arm moves. You will need to position the hands making sure you move the elbow in a way that will sell the idea that the root of the movement is in the shoulders. It is really, really important to pay special attention to the elbow and also the clavicle and chest when you are working in IK. My advice is that you study the movement in question, in your own body and also looking at reference, making sure you understand what is the relationship of shoulder/elbow/wrist for the particular scene you are animating. Then try to get the same relationship going on your shot, even though you are positioning the hands first.
The other important point is to not leave the hands stuck in space while the rest of the body is in movement – a very common error for beginners. That's what we call an “IK-ish look” – for example, let's imagine that a character steps to his right side and reaches out for a glass of water. You will have the shift of weight, the spine/torso moving towards the right, the step, and then the right wrist moves towards the glass of water... but if you forget about the left wrist... the left wrist will be stuck in space as if it is glued to the air. If this happens even for a couple of frames, it will look unnatural; people never have an experience where their whole body moves but their hand gets stuck in mid-air! So, if you have the arms in IK, you have to fight the “IK-ish look” by selling the idea that the movement of the hands is rooted on the shoulders; by making sure the the wrist feels connected to the rest of the body; and by not having the wrists stuck in place while the rest of the body moves around.