The absence of a fan was only part of the problem. (By the way, the Apple II also did not have a fan).  describes it like this:
Those that did work initially often failed after minimal use thanks to Job's insistence that the Apple III not have a fan (a design demand he would make again on the Mac). He reasoned that in addition to reducing radio-frequency emissions (a severe problem with the Apple II), the internal aluminium chassis would conduct heat and keep the delicate components cool. He was wrong.
Compounding the problem was that Jobs dictated the size and shape of the case without concerns for the demands of the electrical engineers, who were then forced to cram boards into a small space with little or no ventilation. As the computer was used, the chips got hot, expanded slightly, and slowly worked their way out of the sockets, at which point the computer simply died.
So we have:
- an internal RF shield impending airflow
- a small space for the motherboard itself, with no airflow
- a higher clock frequency (2 MHz), which makes everything run hotter
- quite a few highly-integrated custom chips (more transistors in a small space = hotter)
- ignoring the engineers
I couldn't find a good picture of the Apple III interior, but if you look at the available ones, you can see a bit how a big part of the motherboard at the bottom is covered by the RF shield.
None of this is a problem in the Altair (or the Apple II): those motherboards don't have custom chips, only standard chips; the clock frequency is lower, and the air can easily circulate above the motherboard vertically between the cards.
 Linzmayer, Owen W. (2004). "Apple III Fiasco". Apple Confidential 2.0: The Definitive History of the World's Most Colorful Company. No Starch Press. p. 42.