Effects of a high SWR.

Audio circuit feedback.
As the transmitter attempts to deal with reflected signal, which it was not designed to cope with, this energy may get into other parts of the transmitter. The effects of this can be squealing, buzzing, warbling and other unwanted noises on your transmitted signal.

RF circuit feedback.
In really bad cases, or where a transmitter has a poor design (many CBs), reflected signal can also cause the transmitter to lock on transmit or to go into self-oscillation.
In this situation, the output is no longer under the control of the internal circuits - and it may reach levels high enough to cook the output and any supply feeding it.
Large supplies have disappeared in an equally large cloud of smoke due to this kind of problem.

Transmitter damage.
Any energy reflected back must be dissipated in some way. That energy which reaches the transmitter may be converted to heat. The transmitter, being designed to be used with the correct antenna, may not be able to get rid of this heat effectively.
The resulting increased temperatures reduce the life span of the semiconductors used in the output stages, and extreme temperatures can instantly destroy such components.
Some components may also be unable to cope with the higher peak voltages created by the combination of the transmitted and reflected wave.

Radiation pattern.
A poorly matched antenna can cause the feeder to radiate, resulting in a change to the radiation pattern. This may not be helpful, as your signal may now be sent into a nearby hill or even up into space.
More about feeder is included in the Feeder sub-section within this site.

Interference.
As any unwanted radiation may be at lower levels, closer to other electronic equipment, interference to other devices is a real possibility.

Movement.
Moving or touching any part of your system may change any of the above effects. Even touching nearby objects, while you have the microphone in your hand, can make the above effects change in some way.
Changes may involve reduction, increase, or additional symptoms occurring.