It is quite obvious that we can't build an amplifier out of resistors (because of their linearity). But is there a circuit theorem that states the impossibility of building an amplifier out of diodes (semiconductor or vacuum tube ones) without using transistors, relays and other 3-terminal devices?
Gunn diodes have a region of negative resistance, i.e., at certain input voltage the current flowing through the diode is increased when the voltage is decreased. These diodes can be used to amplify RF, but they only become practical at microwave frequencies.
You can make an amplifier from tunnel diodes, but that's probably not what you are talking about, since ordinary diodes do not exhibit a negative resistance characteristic, at least not in the normal range of operation.
It's possible to make a parametric amplifier from an ordinary diode because of the capacitance change with voltage. That involves a pump oscillator at a higher frequency than the signal you want to amplify. It has been used for very high frequencies and more moderate ones (eg. Philbrick P2 amplifier).
However, this is an incomplete answer without some kind of reference to a theorem that says the resistive nonlinearity of a diode cannot possibly provide voltage or current gain.