Lets take the example of a very simple first-order crossover. In that case, a capacitor is in series with the signal to the HF driver, and an inductor is in series with the woofer. The capacitor's impedance is high at low frequencies and low at high frequencies. The inductor is the opposite, high impedance at high frequencies and low impedance at low frequencies. The amplifier looks like a voltage source (ideally, with infinite current capability).
The inductor thus "blocks" current flow to the woofer at high frequencies because its impedance is high, and the capacitor does the same at low frequencies to prevent them from reaching the tweeter. The high impedances reduce current flow so there is no effective power out of band (power is voltage times current). An analogy would be a battery with nothing hooked to it; without a load there is no current flow and thus no power dissipation (heat). Ideally no energy is dissipated in the inductor or capacitor, thus no heat in those components. There is stored charge, in a magnetic field in the the inductor and an electric field in the capacitor, but that is ideally lossless. There is parastic resistance etc. in the inductor and capacitor that dissipates energy but those losses are (or should be) very small.
Resistors may be added to attenuate the signal (e.g. to roll off a "hot" tweeter) or aide in impedance matching and they do dissipate energy, but that is usually a very small part of the crossover if even present (many have no resistors).
HTH - Don
p.s. I simplified and ignored parasitic and second-order effects as they do not cause significant dissipation in most designs and I felt the discussion beyond the scope of this post.
"After silence, that which best expresses the inexpressible, is music" - Aldous Huxley