Red shift is the increase in the wavelength of an electromagnetic wave as it travels through space. If the wave travels for a time long enough can its wavelength increase so much that it becomes infinite?If possible, what would its consequences be? The red shift would alter the frequency of the wave, thus would the energy of the wave change?
There are different possible causes for Red Shift, e.g movement of the light source away from the detector (or vice versa), gravitational red shift, or the expansion of space.
For the expansion of space, have a look at the Wikipedia page on Cosmic Microwave Background, in particular...
The photons that existed at the time of photon decoupling have been propagating ever since, though growing fainter and less energetic, since the expansion of space causes their wavelength to increase over time (and wavelength is inversely proportional to energy according to Planck's relation).
So yes the electromagnetic waves will lose energy over time (and their wavelength will increase) but no I don't think it will ever reach zero energy (or infinite wavelength) as this would take infinite time.