Jon:
No, the plate dissipation of a tube based linear RF amplifier run at 500 watts is not greater than when it is run at 1500 watts output - it is normally much less at 500 watts.
That said, the vast majority of linear RF amplifiers I've worked with professionally, both tube and solid-state, are most power efficient at or near 100% output. But in practice, all of them require some degree of "back-off" from full output to maintain signal quality. The degree of back-off usually depends on the transmitted waveform type. But this loss of efficiency does not usually mean that they dissipate more watts as heat, just that a larger percentage of the total power consumed by them is lost to heat.
Amateur amplifiers can be another matter, especially since many are not built to handle maximum output continuously anyway. And SSB in particular has a very high peak to average power ratio, meaning that an SSB signal that reaches 1500 watts peak may only average (for example) 500 watts of RF power. And if the amplifier really maxes out at 1500 watts peak power, it will be very close to becoming non-linear and could easily begin to generate significant distortion products/IMD, and should be "backed off".
It is common practice to tune Amateur tube amplifiers to resonance at reduced power to avoid component over-stress, with tuning touched up at full power, to ensure maximum output power is available on demand.
Brian - K6BRN