Always check to see if your specific model has a service menu that could be used to make whatever adjustments you are needing to make. This menu is often hidden behind a specific button combination.
The largest risk of injury is the large amount of charge that a CRT uses. Even when unplugged, the capacitors in the CRT can give off a nasty shock. It is a dangerous amount of charge and could potentially be fatal depending on how the shock goes through your body and if your heart is weak. Even if you are healthy, a large shock can knock you backwards and cause fall injuries. While some professional CRTs naturally drain off their accumulated charge, you should never rely on that feature to be operational. If possible, work on the CRT with the power OFF and discharge the CRT.
Discharge instructions. Unplug and press power button to turn it to the ON position. Recommended to let sit for a day, then discharge. Sony recommends unplugging the anode cap and touching it to the monitor chassis, but this risks the possibility of dropping the anode cap and touching it to skin.
Remove any rings on your hands that could potentially bridge electrical contacts.
If working on a live CRT, use plastic or wooden tools to perform adjustments to avoid the possibility of bridging electrical contacts with a metal tool.
Optionally use electric "lineman" gloves + leather over-gloves.
The biggest areas to avoid are the anode cap (underneath the suction cup on the back of the tube) and the flyback transformer. Also any AC power circuits are potentially dangerous.
Lifting Safety
Clear the path from start to finish before picking it up. Open doors, make sure no pets can be in the way.
Multiple people if possible.
Use a dolly or cart if available.
Lift with your legs, not your back.
Use non-slip gloves to lift if possible. This will reduce the plastic cutting into your hand and give you a better grip.
Weather Shock Safety
If the CRT set has been out in very cold temperatures, allow it a couple days to warm up inside before powering on. CRTs generate a good amount of heat and coupling that with very cold glass could potentially cause an area of the glass to expand quicker than the rest and break the tube.
Dirty Internals Warning
If it's been exposed to rain and dirt outside, residue can cause bridging between exposed pins and kill the set.
Live near a coast you may have salt build-up that should be cleaned out.
Dust blankets components and causes them to heat up and fail faster.
Always check the weight rating of the platform that you put your CRT set on. Ikea Kallax shelves are a popular choice for displaying multiple game consoles with CRT TV(s) on top, but their weight rating is for downward force and doesn't take into account shear force if something bumps into the side. It's recommended to have back-planes attached to the back or reinforce individual cubbies with metal L brackets.