What is the purpose of diodes D1 and D2 in the 8 Ohm amplifier circuit below?
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In short, those diodes model the forward voltage drop of the BE diodes of Q2 and Q3, so the voltage at the base of Q2 is always two diode voltage drops higher than that on the base of Q3. This is needed for taking over conduction on Q2 as soon Q3 closes and vice versa.
Without those diodes, there would be a gap in the input signal from -0.7V to +0.7V in which none of the transistors would be conducting. Replacing the diodes by a simple resistor has the disadvantage that correction voltage depended on the current then. The diodes have an exponential I/U dependency, so if the current is high enough, the voltage drop on them is nearly constant.
Another measure you could take is mounting the diodes on the same heatsink as the transistors, so temperature difference doesn't change the biasing. But then, better biasing circuits exist und should be used.