There are probably many examples that can be given. First, in my way of thinking, would be any object that has a density less than that of water. In order to measure the volume of the object, the object has to be completely submerged in the water. A piece of a pine log, for example will partially float on the water so the entire object will not be displacing water.
Second would be any object that is made of something that dissolves or that the shape is modified by submersion in water. For example a piece of rock salt would dissolve into the water as the measurement was being made. Objects made of sugar or many other materials would have the same issue.
As other examples of things that cannot have their volumes determined this way: objects that absorb water, a sponge for example, or a stuffed toy or pillow. An ice cube would be another example, as the ice would partially float as well as melt as the measurement was being taken.
I hope this helps you understand the topic.