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When my HVAC system is heating the house, the basement gets super warm and everything in the picture gets warm/hot to touch.
I feel like the system is wasting a chunk of energy. Am I right? Should these pieces be insulated? Or is this part of the design?
FWIW, this is right by the system:
If I haven't missed my guess, this is a temperature sensor. Seems weird to have it so close to a system dumping heat into the room... Maybe not?
Should I insulate or leave it alone?
The "round sensor in the ceiling" is part of a fire alarm and/or ignition/fuel cutoff circuit for the furnace and is EXACTLY where it's supposed to be.
It's a "heat detector" (smoke insensitive, not prone to false triggering) and when things get too hot, it may (depending on what it's connected to) set off an alarm, cut off the fuel supply, or otherwise take action appropriate to having a furnace that is in thermal runaway (or on fire.)
If you want to insulate the rectangular ducts, knock yourself out. If you are getting lot of heat off the round exhaust flue, you might want to contemplate a more efficient furnace; if you want that insulated, you may need to upgrade to (rather expensive) double or triple-wall insulated flue pipe to do the job properly/safely, and then you'll just be conducting even more heat you paid for outside than you are now.
As isherwood mentioned in a comment, that round pipe is the flue. It's what takes the hot, toxic, exhaust gasses from the burned fuel and gets them out of your house so you don't asphyxiate and die.
You can insulate the flue with an appropriate high temperature insulation and ensure that all the heat left in the flue gasses is piped directly out of your house and into the atmosphere, or, you could let more heat radiate from the flue into the house and use that heat (even if less efficiently) within the insulated envelope of your house. I'd open the basement door to let all that heat escape up the stairs and help heat the first floor.
Your call.
As for the sensor on the ceiling. Is that a temperature sensor or is it a CO detector?
If it is a thermal sensor, that does seem to be an unusual location for it, but it may have been that a previous owner found that the furnace room was getting even warmer than it is now, and installed that as a backup cut-off switch.
leave it alone and install a vent pass through in your basement door to allow the hot air into the rest of the house. much cheaper solution than the heat resistant insulation. you can even get powered ones if you are willing to put it in the floor somewhere.
