Conservation of energy and Doppler effect?

Conservation of energy doesn't apply to this situation because the energy you measure when at rest with respect to the source and the energy you measure when moving with respect to the source are in different reference frames. Energy is not conserved between different reference frames, in the sense that if you measure an amount of energy in one reference frame, and you measure the corresponding amount of energy in a different reference frame, the conservation law tells you nothing about whether those two measured values should be the same or different. If you're going to use conservation of energy, you have to make all your measurements without changing velocity.

In fact, it's kind of misleading to say that energy increases or decreases due to a Doppler shift, because that would imply that there is some physical process changing the energy of the photon. That's really not the case here, it's simply that energy is a quantity for which the value you measure depends on how you measure it.

For more information, have a look at Is kinetic energy a relative quantity? Will it make inconsistent equations when applying it to the conservation of energy equations?.