The fact that all your wires are red does not matter too much as you have no rear door bell button to worry about.
Normally, power would start at the transformer, go to the TRANS connector on the chime via one red wire, from FRONT go to the front door via another red wire, and then come back from the button to the transformer via the third red wire. The button is normally open; you push the button, current flows through the button and the chime rings. If you hold the button closed, the chime will ring once and then sit there, humming and overheating until someone comes and forcefully takes your finger off the button.
So if the button is removed, but connecting the two red wires at the chime causes a muted ding, continued buzzing, and heat, that suggests that something similar to the button being pressed and held has happened. The most likely thing is that the two red wires that go to the button, one from the transformer and one from the FRONT screw on the chime, have somehow gotten connected to each other by something other than the button - if I was to guess, most likely I'd say they are held in place with staples, and one of the staples has now made contact with both wires. Possibly rodents have eaten the insulation, they seem to love doing that. Almost certainly you won't be able to find where this has happened, but you can follow the wires, if it's an older house, and look for it. (Newer houses seem to be constructed so as to hide all their inner workings. My son has just bought his first house, and had a truly annoying time trying to pull a network cable because of that.)
If you can find where bare copper from the two wires is touching, and fix that with (e.g.) electrical tape, you can then hook everything up again and have a working door chime. If you can find the full run of wire, you can simply leave the old stuff in place and pull new. Otherwise, I'd go with the suggestion of getting a wireless doorbell, and I'd also suggest that you change the battery in the doorbell button once a year, at the same time you do your smoke detectors.
Edit: If you want to follow and try to repair your existing wires, you need to follow the wire from the chime to the door; and almost certainly, the wire from the transformer to the door goes past the chime, but doesn't surface at the chime, so you can't immediately see it. Unscrew the chime from the wall, look through the hole the wires come out of, likely you'll see the third wire through that hole. You may also be able to see the direction the wires take after they go by the chime. If they go up, likely they are routed through your ceiling to the front door. If they go down, they may surface somewhere else in the basement and run towards the front door from there. But unless there is only attic above your ceiling, and you are confident about walking around up there, if the wires go into your ceiling it's unlikely you'll be able to find and repair them. If you are able and willing to go into that space, the wires are likely stapled to the joists - the beams that hold the ceiling up. It's at the staples where I'd expect to see wear and bare copper.