Do loose connections use excess energy ? Do they waste electricity ?

Only if they result in heat. If the wire is loose but there is no heating, there is no waste.

My thinking is that a "loose connection" equals either an open circuit or a higher resitance joint. With no current flow, no real power is "wasted". With current flow, you have higher I^2*R (watts) losses, so technically it's "wasted". But obviously, the real issue is that this energy can be enough to cause heating (and arcing), which leads to bad outcomes.

Since loose connections usually get hot they waste power. Many electrical caused fires start with a loose connection. The longest time from connection to fire I have seen was a house wired circa 1910 had a connection related fire around 2000 or 2001. It took 90 years to ignite the structure and still left enough evidence to reveal the cause. That was a cold solder joint in Knob and tube.

The amount of energy in a hot bad connection may be similar to that used by a 30W soldering iron. It won't run up the electric bill much, but that energy is happening where you don't want it.

OK, now it's math time, based on Wa2ise's input.

30 watts at 24 hrs equals 720 watts. Seven days a week = 5040 watts, carry on to a 52 week year, and we have 262 KWHr. Using approx. $ 0.15/KWHr. = $ 39.31 cost

Now....that's only one 'loose connection.

Further, using Mikesh's comment of approx. 90 years...$ 3538.00 wasted, excluding the replacement cost for the fire damage. I'll need one of those actuary type bean counters to factor in the cost of electrical energy for that 90 year span, thou.

This becomes a lot more complicated when you put the voltage drop in there. The entire circuit will have higher resistance so the equipment plus the bad splice uses less power than it would if the splice was OK but voltage drop may make motors less efficient and you lose that saving. If it was only lights and you were happy with the light output it is better than a wash. You save money on power and your bulbs last longer. The extra heat costs money in the summer when the A/C is on, again probably a wash when the heat is on.

All of that is true until the fire starts

Greg:

I didn't factor in anything except the hypothetical soldering iron 30 watt figure.

I give my students a typical 'line-loss' calc with the Vd calcs. Boy, you want to see some $$$ costs from the math challenged!

Just bear in mind utilities "brown us out" when they want to reduce consumption. All of the I2R calculations I see assume the load will still be drawing the same current with the dropped voltage.
Pretty soon we will be revisiting the light bulb saver thread.