We have investigated the impact of surface recombination on the effective carrier injection efficiency and the Joule heating of axial InGaN/GaN nanowire light-emitting diodes (LEDs). The results reveal that the carrier injection efficiency of such devices is extremely low (<10%), due to the severe carrier loss through nonradiative surface recombination. It is further observed that the thermal resistance of typical nanowire LEDs is comparable with, or lower than that of their planar counterparts, in spite of the reduced thermal conductivity of nanowires. The poor carrier injection efficiency, however, leads to significantly elevated junction temperatures for nanowire LEDs. We have further demonstrated, both theoretically and experimentally, that the carrier injection efficiency can be significantly improved in p-doped nanowires, due to the downward surface band bending, and in InGaN/GaN/AlGaN dot-in-a-wire core-shell nanoscale heterostructures, due to the superior carrier confinement offered by the large bandgap AlGaN shell. This paper offers important insight for the design and epitaxial growth of high-performance nanowire LEDs.