If a force is performing work, then it is causing some kind of change in the thing receiving the force: position, velocity, temperature, shape, etc. If a force is causing no changes, then it is doing no work (see the answer here).
In the case of the table, placing any weight on top will cause it to deform a like a spring. After this initial deformation (which does require work to be done on the table, since it changed), there are two possible end results. First, the weight is light enough that the table can hold it up indefinitely without further changes in shape. Second, the weight is heavy enough that the table continues to deform and sag to the ground before breaking (which may be a fast or slow process). The first situation is called elastic deformation since the table will return to its original shape one the weight is removed. The second situation is called inelastic deformation. Removing the weight will no return the table to its original shape (especially if it's broken).
Whether the deformation is elastic or inelastic comes down to whether the weight is large enough to overcome the forces holding the table together, namely the bonds between the atoms that make up the table. Elastic deformation only stresses and stretches these bonds. Inelastic deformations break these bonds, resulting in permanent changes to the structure of the table. If the table continuously sags due to the weight placed on it, that is evidence that the structure of the table is changing, and that the force is doing work on the table. If the table is holding steady, then no work is being done because nothing is changing.
In the case of your arm, as described in this answer, the structures that cause your arm to hold up a weight are temporary by nature. Thus, to hold up a weight, the structures that cause your muscles to contract have to be constantly reformed. It's a little difficult to tell that work is happening because your arm is mostly motionless. However, if you were to look at your arm with an infrared camera, it would be noticeably hotter than the rest of your body. Since heat radiates away, maintaining an elevated temperature requires work. So, your arm is performing work, but the work performed goes into heating your arm. No work goes into the weight because the weight is not changing: not in position, velocity, shape, nor temperature.