I am looking at replacing my doorbell (original to the house, circa 1978) with something more modern looking. Now, the problem I have is that most of the modern doorbells are wireless. Mine is wired...and I can't find any information about terminating the power to the doorbell and ideally burying it in the wall to be forgotten about. Is this possible? Or if it isn't what are feasible, ethical and non-burn my house down options?
Don‘t tear the wires out
Some newer doorbells are going wireless. But this is not a universal trend. Better doorbells are actually wired, because they do things that take power. They are lighted (like, the button glows), they have cameras or WiFi, motion sensors, microphones, speakers, etc. Those things are not possible on battery.
Also, the wireless doorbells I have seen are not well built, and don't last. I would expect quality ones to work both wired or wireless.
It would be silly to, a few years later, pay an electrician to re-run the wires... Or have to forego a nice doorbell because of no wires.
In other words, I think "wireless" is a fad.
I would simply disconnect the wires at both ends without cutting or damaging them. The other end will be the chime, and it's quite likely the transformer is near that. Sometimes, doorbells coattail on the furnace/thermostat transformer.
Those wires going to your doorbell button hook up to a transformer somewhere. Maybe it's behind your chime, or maybe its in the attic, but it's there somewhere. Find the transformer and completely disconnect it (but leave it there in case anyone wants to hook it up again), and then you can bury the doorbell wires in the wall. Again, try to make the wires accessible even if you tie a string onto them in case someone else needs it later.
If you can't find the transformer, you can cap off the doorbell wires with wire nuts, but that transformer is going to still be live in the house somewhere using a small amount of power. Not really recommended.
There are "wireless" doorbells that can actually take power from the low voltage doorbell wiring, though you may need to adjust the wiring to make it a power delivery circuit instead of a "short-circuit to ring" circuit.
These have the advantage that you don't have to worry about charging the battery every so often.
To power the wired doorbell, there will be a transformer that converts the house voltage to a lower voltage. To decomission the wiring you'll have to find that transformer and remove it.
The other option would be to cap them off and leave them inside the box of the new doorbell. That lets you or a future homeowner use them in the future.