Will three phase power damage single phase appliances?

If you plug a single phase appliance into a three phase power supply, then you will be using single phase power. It does not matter if you are delta connected, AB, BC, or CA, or if you are wye connected, AN, BN, or CN. Yes, if there was "a problem", you would trip the supply. Question, however, is what do you define as "a problem"? Three phase power is intended to supply three phase appliances. Connecting a single phase appliance to three phase power is inconsistent with the objective, and such connection must have been performed by some kind of "jury-rigging". If you pull more than the trip current on any one phase, the supply should trip. If a malfunction in the single phase application, however, were to result in fault current that is lower than the trip current, you might not trip, and you might create damage and/or a fire. There is nothing wrong with building a branch circuit that feed a single phase load from one phase of a three phase source. You just need to provide the correct protection for that intended load.