Output voltage of LC circuit?
The LC circuit exists to select the AM station in this corcuit here. These devices are intended to be connected to an antenna large enough to pick up significant AM signal (there is no amplification, the only power source is the received signal itself). Also the earpiece should be very high impedance in order to efficiently convert the tiny signal into sound.
You can't really know from the diagram what the voltage out is. It's just a filtered (frequency selected) version of the signal picked up on the antenna. Without it, you should hear wideband noise, with any transmitted signals probably lost in the noise.
The voltage across the speaker is just the voltage available on the (loaded) antenna (which depends on received signal strength and the antenna itself), half-rectified, filtered according to the characteristics of the tank circuit, and less a diode drop.