Not having made cottage cheese before, I'm unsure of what effect it has on the quantity produced. However, I do know that it has a significant effect on the flavor of the finished product. Just like skim milk tastes blander than whole milk, the same applies to cheese. When you buy non/low-fat cottage cheese in the supermarket you'll notice they add sugar to prevent it from tasting like runny mashed styrofoam.
Update - I sent some emails out to a few food science professors regarding the role of fat in cheese yield, and got a response from Art Hill, Professor and Chair, Department of Food Science, University of Guelph.
Fat is a principal yield component in cheese. Fat is trapped in the casein protein matrix during cheese making. Thus the mass of the fat contributes to the cheese yield. However, lower fat cheese often contains more water so yield loss is partially compensated by increased water retention.
I did some additional research and found that many cheeses have strict MNFS (moisture in the non-fat substance, calculated by subtracting the fat and expressing the moisture as a percentage of what is left) limits in which they can legally be called whatever they claim to be. Take Cheddar for example, the highest moisture, lowest fat cheese that can legally be called Cheddar 56.12% MNFS. To limit the moisture content of low-fat cheddar at less than a 1/3 fat reduction, fat substitutes are used. These can include protein based beads designed to mimic fat globules, and starches.