Solid state for the most part does not care what the load is nor does it care if there's no load attached.
Tube amps can be finicky. As a very general statement, OTL tube amps want to see a specific load for optimum performance. Some transformer coupled amps may have specific desired loads as well. Sometimes, these "dummy loads" are built directly into the amp. But sometimes not. Transformer coupled amps also generally do not like to run without any load attached, and may also have dummy loads built in, but sometimes not.
In the speaker world, usually you attach the speakers to the amp and leave it there and that's it. So even if you have a transformer coupled tube amp, chances are you never run it without a load so there isn't too much to worry about. With headphones though, we have this annoying habit of unplugging the headphones which brings the risk of running the amp without a load which can potentially blow the transformers or the tubes or both.
So thus the quick fix is to slap a resistor on the outputs. Most safe bets say to put a 10ohm resistor (of at least 5 or 10W rating) across the outputs of each channel. Just this by itself doesn't offer any attenuation though.
The other popular arrangement is the L-pad. Google it for a dozen websites that tell you how to build one with built-in calculators and schematics. Basically you use two resistors to create a voltage divider for your headphones, If you pick the resistors carefully, you can also get a decent damping factor and come close to the desired amp load if required. Sometimes getting all that with two resistors is tricky though, so a three resistor arrangement might be used.
And then there's my stance on the whole thing: don't pick a huge amp that necessitates those resistors to begin with. If there's too much gain on the amp or you have noise issues, pick a different amp.