All components have a rated life that includes the life at different operating temperatures. It is always the case that higher heat means shorter life. If an amp is run constantly, mostly at idle, I would hope that it is designed such that heat would not be excessive, but, in any case, that heat and constant use would not increase the remaining life. It is particularly the case that heat as well as constant operation would not be good for transistors. As for thermal shock affecting PCBs, I have my doubts that normal turn on and off operation would not have a meaningful impact on life of boards, and would be interested in information demonstrating otherwise. PCBs are tested for thermal shock vulnerability, but testing conditions are quite extreme (like -40 centigrade to 100 centigrade), not the kind of conditions found in home operation. Boards are expected to last for at least 50 years.
Electrolytic caps do sound their best and are well preserved by being kept charged, so there may be some benefit there to keeping an amp on all of the time. But, they too are subject to shorter life from higher temperatures, so, some sort of tradeoff/balancing is involved there. The big issue with leaving electrolytic caps in a discharged state involves many years of storage in such state--it is not really an issue as far as a few days or even months of non-use. But, sound quality is another matter. Some gear takes a long time to sound its best even when warmed to full operating temperature because the caps are not fully charged/formed. For example, NAIM recommends leaving their streamer on all of the time because it takes the caps almost a whole day to be at their best from a cold start.
The other safety/reliability issue has to do with power surges from electrical storms and brownouts that may occur while one is not at home and able to turn off gear. I have not heard of many instances where gear has failed and caused a fire, but, that too is a (remote) possibility.
When I ran solid state amps, I kept them on all of the time, except when I left town. I knew this involved some risk and shortening of the life of the amps, but, I accepted that to avoid a warm up period. I now run tube gear, which MUST be shut down when not in use to preserve tube life, but then again, tube gear tends to sound quite decent after a relatively short warm up period. I can enjoy the sound, even if it might not be at its very best, within five minutes of turning on the amp and linestage.