Examples: A compressed spring weighs more than the same spring in an uncompressed state. A box of light (e.g., a box with perfectly reflective mirrored walls) weighs more than the same box without light, despite the fact that photons have zero rest mass. The bulk of a proton's "effective mass" is due to the kinetic energy of the quarks that comprise it. The joint earth + moon system weighs less than if you added up each component weighed in isolation.
The other interesting fact is that the Heisenberg Uncertainty Principle is commonly misunderstood to mean that you can't simultaneously measure the position and momentum of a particle. In fact, you can indeed obtain partial information on both properties at the same time, and there's quite a few papers out there describing how to perform joint measurements of incompatible observables.
What HUP more accurately entails is that, for a quantum state corresponding to a specific system, you cannot obtain complete information about the two properties — no matter what you do. This is because performing a position measurement destroys your ability to perform a subsequent momentum measurement (on the same system) and vice versa. One of the postulates of quantum mechanics is that measuring a system collapses it into an eigenstate of the observable corresponding to the type of measurement that was performed. Since the position and momentum operators don't commute, it's not possible to put the two into a joint eigenbasis (they can't be simultaneously diagonalized).