Do quantum particles have mass?
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I think it would be helpful for you to review the wikipedia page on particle-wave duality. If you have access to a textbook on the origins of quantum mechanics, that would help clear things up,too.

The quantum mechanical model for particles states they have both particle-like and wave-like behavior, so it is incorrect to say they are never particles, which is what I assume you mean by "objects".

At the most fundamental level, quantum mechanics is just another tool with which to view and understand the world around us. It's an especially useful tool when we are looking at things in very fine detail (let's say, usually &lt; $10^{-7}$ m). So, we can look at a given particle from the framework of classical mechanics or from the framework of quantum mechanics. This is an important point, because it is not quantum or classical mechanics that attributes mass to the particle. Rather, the particle had that mass to begin with. 

So, to answer your question: A quantum mechanical treatment neither prescribes nor removes mass from a particle or an object. A photon is massless, a proton has mass, and that's that!