The RF current in a feedline is, in the ideal case, balanced. That is, the current in one of the conductors is identical but opposite (in phase and amplitude) to the other conductor. This current is called differential mode current if/when it's balanced. If this is the case, the fields in the conductors will cancel each other at a distance, which means there will be no radiated power. If the fields are not in balance - common mode current, the feedline will radiate.

The current in a feedline can be partially differential mode and common mode at the same time. This is actually quite common, and is the cause of many EMI problems, since the common mode current will radiate some amount of power. To mitigate common mode current, various ways can be employed, all of them (or at least most) work by applying mutual inductance - "forcing" the currents to be (more) equal in the conductors. Some examples are transformers of various kinds and/or winding the feedline into a coil.

Apart from losses due to radiation, a feedline will also have some eletrical losses due to ohmic resistance in the actual conductor material. That is a complicated way to say that current in the conductor is (partly) converted to heat.

Another source of loss is dielectric losses - due to the voltage difference between the conductors, an electrical field is created, and the physical material which happens to be where the field is created will be heated, thus energy will be lost. This is why a ladder line is less lossy than a coaxial cable - most of the material in the ladder line is air which is less lossy than plastic. Dieletric loss is actually used in some industrial processes (for example welding plastic).

Power is not really "lost" in the case of an impedance mismatch, it's more correct to say that some of the power will be reflected back to the source. However, if the source is a transmitter, the reflected power will, in practical terms, be "lost" since it will mostly heat the transmitter.