From a purely economic perspective, the Holocaust was a terrible allocation of human resources within the economy. While some of the people sent to the camps could potentially have provided value as manual laborers -- particularly those forced into labor that no one would voluntarily do, such as human experimentation subjects or perilous chemical work -- many were possessed of skills and knowledge that would have been much better allocated to specialized areas that were more valuable to the German war effort. This misallocation of resources, that is, causing resources to be used in a situation where they are not at primary efficiency, is known as opportunity cost, and is one of the most critical concepts in economic theory.

The most obvious better use the Germans could have gotten out of the Jews and other concentration camp victims would have been as soldiers. Millions of able-bodied soldiers were sacrificed this way, and thousands more were made to "guard" them. It is hard to conceive of a historical situation in which the labor that went on in most concentration camps was of more value to the Germans, if performed by an able-bodied male of fighting age, than soldiering, and certainly nothing could have been more valuable than potentially skilled pilots.

Beyond this, however, many of the Jews and others who were sent to the camps were highly educated and accomplished in various important fields, such as engineering, rocketry, chemistry, physics, ballistics, and many other fields. The loss of these certainly set back many German efforts in military science and other fields, and cannot possibly have been compensated for by any manual labor received in return. Writers, historians, artists, and others, could have had at the very least a powerful propaganda impact, and possibly could have had other morale effects, as well as potential as translators and journalists, also not compensable by manual labor.

In short, it is difficult to see how the positives could outweigh the negatives in terms of opportunity cost of the Holocaust, whatever the Germans got out of it.

That said, it's important to understand that the purpose of the concentration camps was not, primarily to derive forced labor. The goal the Germans had in mind was the eventual extermination (after prolonged suffering) of the interned. To that end, it doesn't really matter what benefits, if any, the Germans received from the concentration camp labor. The real "benefit" was the extermination of the camp detainees, which was largely achieved.

Sources:

Modernity and the Holocaust, Bauman, Zygmunt

The Economic History of the Jewish People, Attali, Jacques

There was acute labor shortage that Nazi Germany experienced mostly due to mobilization. It tried to alleviate it by importing millions of forced foreign laborers, but it was not part of the concentration camp system (even thought it was effectively slave labor), which had other purpose, and was net economic drain, even if sometimes these conicided like in the famous Auschwitz chemical plants.

I am not sure if anyone knows the specific answer, since the matter is quite complicated (so many different resources going into the holocaust business), and answering it would be a research project in itself.