I'm hampered by not seeing all of the connections, water and electrical, but it appears to be the thermostatic control to a small pump that will circulate your domestic hot water through a loop in order to give you instant hot water at your faucets.
JMHO

That barrel-shaped red thing below it is probably the pump. I'm sure there used to be a cover. Those are likely live electrical lines. The dial is a thermostat.

That control is an Aquastat which measures the actual temperature of the return water in your loop and it only turns the pump on when the return water temp falls below a preset point.

These are actually much more fuel efficient than the typical timer method that is commonly used. With a timer the pump is turned on every 15 minutes or so and runs a couple minutes to insure the water in the loop is hot, but with an aquastat it measures the temp of the water in the return loop and only turns the pump on when the temperature drops below the set point. Your hot water & return loop lines are well insulated (which is required by code on a loop)so they standby heat loss from the lines is minimized however the rate of heat loss varies by season. Needless to say, the lines would cool faster in winter when your basement is cooler than they would in summer. The aquastat can automatically compensate for seasonal change simply by measuring the actual water temps.

For peak fuel economy an aquastat can be put in series with a timer. In this manner the aquastate would keep the lines warm during the active part of your day, but a timer could be set to turn it off during the night when there is little likelihood of a demand for hot water.

That cylindrical red device that is connected to the water lines with flanges is your circulating pump.

The system requires a very small amount of water to keep the lines full of hot water. By example, a half inch line holds one gallon for each 98 linear feet of pipe.

Your inspector is correct, there are some bare electrical contacts inside that aquastat enclosure that really should be covered to prevent someone from touching them inadvertantly, but there is a simple solution.

First, look around on the floor very carefully. In many cases someone has removed the cover and laid it on the floor, then got in a hurry and didn't replace the cover. If you should happen to find it, you can just slip it back on.

If you can't find the cover you could drive yourself crazy running around from store to store trying to find one, but that would soon prove to be a lesson in futility. The simple solution would be to just get a small piece of sheet metal and form a new cover. I am attaching a drawing with details on how to make it. If you don't want to make it, you could measure the dimensions of your aquastat and take that to a sheet metal man and he/she could knock you out one in 5 minutes.

You will need to determine the size of the screw required to hold the cover on and if you don't have one in a junk drawer you could get one at any hardware for about $.10