The efficiency will be lower as the antenna is operated off resonance.  There is a common misunderstanding in relation to operating at resonance and feed point mismatch and that a good match is what's needed to get good efficiency - not true.

Adjust the antenna for operation at resonance then adjust the matching for a good match for best efficiency.

Mirok,

this is a good question. The radiation efficiency has nothing whatsoever to do with resonance as such. The radiation efficiency is the ratio of power radiated to power accepted by the antenna. It is given in antenna text books by the ratio of the radiation resistance to the total resistance. The total resistance is the radiation resistance plus the loss resistance.

So, it should be clear that to get the best (theoretical) efficiency you simply need to maximise this ratio of resistances. Now in practice if the reactive part of the antenna’s impedance gets very high the theoretical "lossless matching network" used to match the antenna to the feedline and amplifier will give losses that need to be considered when calculating the overall system efficiency. Thus it is highly desirable to have the antenna as close to the characteristic impedance of the system as possible, minimising the amount of matching needed.

In practice the reactance of an antenna can increase so rapidly away from resonance that the overall system efficiency will be low if the antenna is run off resonance.

logbook, I agree with your coments completely and have noted many occasions where antennas were designed so that the feed point impedance was as close to the characteristic impedance as possible.  Commonly this was achieved by techniques that really increased the loss resistance, adjust Rl so that Rr+Rl=Zo.  This is why I advocate the principle of adjusting the antenna for operation at resonance then matching it to Zo.  For a given antenna reducing Rl to zero will lower the feed impedance and increase the matching task, but you live with that.  

Playing with the feed point can give a match closer to Zo but as you suggested it is generally highly reactive and will have a narrow bandwidth.

This probably wouldn't apply to a typical microstrip (GHz)  antenna, but low efficiency can be caused by a short (fractional wavelength) antenna where the feedpoint Z is very low.  The very low Z, even when matched, causes high currents and IR losses (thus poor efficiency).  This is typically a problem with practical mobile (small) antennas at the low end of the HF band (or lower) when transmitting.

In the case mentioned above, off resonance might make it difficult to feed the antenna, but reflected power isn't quite the same problem as (the proper meaning of) low efficiency (except that the matching network will cause additional loss).