My gas furnace (Trane XL80) this morning started to smell like plastic was burning from the registers, I went to check the furnace and noticed that the blower wasn't working so I turned the breaker off. My question is, should the burner had to turned off if the blower wasn't working for some time?

Sounds like a failed blower motor. Eventually the high-limit control would have shut down the burners, but not until the limit temperature was reached (typically 160-220 degrees at the limit control location). Try this: Set thermostat system selector switch to "off" and fan selector to "on". Does blower run? If so, allow it to run for a while, see if the motor overheats and shuts down.

The blower did not work when I did the changes on the thermostat.

I will change the blower and capacitor.

Thanks for the help.

Your welcome. P.S. Carefully feel the motor. If it's really hot to the touch, it has overheated and activated the internal thermal protection device. That would explain why it doesn't run now.