Tubes are biased at a certain voltage to get the optimal performance according to its design.
Without getting too complicated, tubes have to run at a certain voltage, and then the voltage goes up and down depending on the signal.
A tube cannot go negative. If the bias was zero, it won't go to, say, -25V when the signal dips that low. You have to set it (for example) at 200V. That way, when the signal dips -25V, it'll go to 175V, and also swing back up to 225V in the other direction.
If you look at tube curve charts, you'll see that a tube can be biased at a variety of voltages. Each voltage produces a slightly different curve - some more linear than others. Where the amp's designer sets the bias changes how the tube will operate in the circuit.
Now, as the tube changes with age and use, the bias point also changes. On some amps, you have to turn a knob connected to a potentiometer that changes the voltage going into the tube so it keeps operating at the optimal bias point selected by the designer.
Other amps, however, are designed so the circuit biases itself with tube age, to keep it operating where it should.
As has been pointed out, most headphone amps do not require this bias adjustment. I can't think of a commercial model that requires you to bias it. Even if you find an amp that requires biasing, it's not a big deal.
My Conrad-Johnson MV52 (speaker amp) has to be biased. All I have to do is turn a screw until a LED turns on. Then I back it off slowly until the LED goes out. It takes about five minutes and is as easy as it sounds.
As for other maintenance, there ain't much. Don't leave it on when you're not using it. Tubes aren't the only components that age with use. Capacitors and resistors have a finite lifespan, too, and you don't want to wear them out before their time. Heat is the enemy of all electronics - the more heat they're exposed to, the shorter their lifespan. That, and you'll eventually have to change tubes. It's about as tough as changing a lightbulb.
So don't worry and enjoy your tubes.