The problem here is that there are no calories going into the system. Plants extract calories from the sun, but fungi extract calories from whatever they live on. The mushrooms must have fewer calories worth of food than the feces that went in, because the fungi use the calories to live and grow. There would rapidly be less calories for humans, less feces for mushrooms, and the system would quickly grind to a halt.
Mushrooms are also low in calories and typically vitamins. They can or do contain vitamin D, Zinc, and potassium. They are an incomplete and generally somewhat low source of protein. Here is where genetic engineering would come in handy. You can add the enzymes and pathways to allow your fungi to produce more and more varieties of amino acids to provide complete protein. You might even be able to make the fungus out of a digestible starch like a plant, but that would vastly increase the caloric needs of the fungus and only exacerbate your calorie problem. Raising the fungi with more minerals and the engineered ability to absorb and sequester them Could make the mineral content balanced and complete. But the fungi would still need a source for those minerals.
Heat from outside is an extremely poor energy source. Besides the fact that fungi can't use heat for energy, heat is already an extremely disorganized state of energy. Unless you have an epic source of heat (like the sun) it will be the lowest energy state entropically.
- As a side note, plants provide oxygen, so I hope your colony has an external source of oxygen. They also fix carbon, which will be decreasing as people and fungi respire CO2.
But wait, there is SOME hope.
If you have geothermal heat or any other source providing electrical power, and are growing something photosynthetic and inedible (like an algae) in tanks using light, this photosynthesizer can be fed to the mushrooms, the mushrooms will have an energy input (rotting the algae) and this becomes possible.
Based on the question, I assume just growing plants for food is counter to the goals of the question.
If your colony has an external source of biomass (like alien plants, if it is on another world), a fungus could be engineered to break down the alien sugars and amino acids (or equivalents) and use these to produce food humans could eat. I've used something similar in some of my stories.
In the somewhat unlikely category, you could have a non-renewable supply of energy. If you have a fungus that can utilize some chemical in the rocks for energy, then you have a (non-renewable) source of energy. Then you would need to provide a stream of the mineral to the fungus, or constantly expand the fungus to new deposits. If the fungus can use petroleum, and you have access to oil or tar pits, the fungus could consume those for energy. But for this kind of activity, bacteria usually do the job of utilizing exotic chemical sources of energy better. You might need something more akin to a lichen (but with bacteria as the symbiote) for your fungus.