It sounds like there's an ungrounded (hot) conductor shorted to ground somewhere. Finding the short is likely going to be a tedious task.
Check your work
Did you install grommets or clamps, where the wiring passed through knockout holes in boxes? If not, start by inspecting the wiring where it passes through the knockouts. The edges of these holes (especially on thin recessed lighting boxes) can be quite sharp, and can easily damage cables and wires as they pass through the hole.
Check all your connections, making sure nothing came into contact with ground or non-current carrying metal. This can sometimes happen when wires are pushed into boxes, so check for loose or pulled out wires.
Inspect cables/wires where they are supported, especially where supported by metal staples or clips.
Check existing work
Start by turning off each breaker in the panel one at a time, while monitoring the voltage on the drop ceiling rails with your non-contact voltage tester. When the voltage goes away, you've found the offending circuit.
Once you know which circuit is causing the problem, it's time to start the tedious task of locating the fault. Do your best to trace out the circuit, and determine all the equipment on it. Open up all outlets and equipment, and inspect the wiring inside.
It may help to move along the circuit, and disconnect the load conductors (the conductors feeding the next junction) at each junction. Then energize the circuit, and check for the voltage again. This can help you isolate the problem, and will help you focus on the problem wiring. Once you find the offending part of the circuit, methodically inspect the wiring. Checking for damaged wiring, loose connections, improper wiring, etc.
Unless you get really lucky, you're likely in for a long tedious search. Good luck.
The wire wrapped around the screw and the wire "stabbed" into the port next to it are connected together internally within the switch. With the switch off, test for voltage to determine if the lone wire, on the bottom of the switch, is the hot (line) or if the two connected wires, on the top, are together the hot (line). The two wires currently connected together on your existing switch should be again connected together on your new switch.
Good Luck,
Volt-Ron
Best Answer
That's a hot, not a ground
What you're seeing is an always-hot wire most likely, not a ground, that lands on the terminal screw then continues onward somewhere else in the circuit; some electricians will cut the wire and pigtail, others will use both terminals in a screw-and-clamp, and yet others will wrap a stripped section mid-wire around the screw as you see here. The 3-way Kasas, fortunately, have terminal screws, so you can leave it "as is", or pigtail it; if you were using a switch with a wire pigtail built-in instead, you'd have to cut and strip so that you could nut the pigtail to the cut ends.
Good news: this is a conduit job
The yellow wires and the use of a metal box with a mudring also tell me that this is a conduit job, which is good news: it not only means that you likely have metal conduit providing your grounding path, it means that even if there's not a neutral already in this box, pulling a neutral to this box is an easy job for any electrician. It also means that you don't need a grounding wire for the switch; instead, it simply can pick up ground via its mounting screws.
Good news #2: neutral appears to be present
The other good news is that since this is a conduit job, we know that white wires must be neutral in order to meet Code. So, we know offhand that those two bundled white wires must be a neutral bundle. However, since one conduit may have multiple circuits in it, we'll need to know more about the inside of the box in order to figure out if it's the correct neutral for you to connect to.
Bad news: this is a 3-way
The bad news about that pair of yellow wires connected to separate switch terminals, though, is that they tell us this is one end of a 3-way switch arrangement. This means that you'll need the 3-way version of the Kasa smart switch, part number HS210, instead of a single pole switch, and will need to find the other 3-way in the complex so that you can replace it as well.