Old tires can be placed into a dumpster if there's no other tire recycling services near you.
The trash companies now in most places sort through the trash that goes into the dumpster so the tires would get removed at the trash and recycling facility.
We have a public trash dumpster here in town and I have placed tires into the dumpster to get rid of them and they've always took them.
Even though the dumpster says no tires I still put them in there because I have no other place to take them.
However some places might take them such as tire places for a small tire disposal fee or your landfill may have a place to take tires for recycling.
In some places you may be able to sell tires as well.
But not all places have places or companies that buy used tires for scrap but some do.
Some towns will also hold a tire pickup service once per year where you can drop your tires off or sit them at the curb and they will usually be picked up.