Would the portable ~$40 Tecsun get damaged somehow by a 100-foot antenna because it's designed to work with shorter, portable antennas?

Unless you are within a kilometer or two of a very high power transmitter, there is no chance of your antenna collecting enough power damage your receiver.

You are more likely to cause damage to your receiver from static build up on the antenna or nearby lightning strikes inducing too much power into your receiver's front end. Disconnecting the antenna and shorting it to ground when not in use and adding a basic lightning protection circuit will mitigate this potential.

Would the portable ~$40 Tecsun get damaged somehow by a 100-foot antenna because it's designed to work with shorter, portable antennas?

Unless you are within a kilometer or two of a very high power transmitter, there is no chance of your antenna collecting enough power damage your receiver.

You are more likely to cause damage to your receiver from static build up on the antenna or nearby lightning strikes inducing too much power into your receiver's front end. Disconnecting the antenna and shorting it to ground when not in use and adding a basic lightning protection circuit will mitigate this potential. On a receive only antenna, another technique to bleed off potentially damaging static build up is to connect a 5-10 meg ohm, non-inductive resistor between the receiver end of the antenna and ground.