I assume you are talking about these:
http://www.jacksonampworks.com/images/at_preamp_tubes.jpg
They are shields. Implying that they shield the tubes from picking up any outside signals (AC fields, RF signals, etc)
How important are they? It depends on the circuit they are used in.
When employed in audio circuits, they are almost always on tubes in high-gain stages like phono, guitar or mike preamps, to prevent the stages from picking up hum or other spurious signals, and to prevent the stages from going into oscillation. In lower gain circuits such as phase inverters, they are not required in most cases.
In radios, TVs and tuners they are used to prevent tubes in different stages from interacting with each other (as there may be signals radiating from the tubes in some cases (Oscillator stage in a TV or FM tuner for example)), which would cause problems with the proper operation of the equipment.
They aren't dampers or heatsinks. (if they were used as heatsinks, you'd see power tubes with them all the time. You don't for a reason).
-Aaron
