heater current at 6.3V *.3A per tube with triodes connected in series is ~1.89W - not enough to make a drastic difference as far as heat is concerned. the original intent of the metal casings is RF/EM interference - hence Jerry's reason in the other thread about putting the cover back on jake's 5150 as a possible solution.
however, this is to show that it wont cause a tube to overheat. this does not mean that it will not have any effects on sound. being as electron flow is what matters here in this variable i can guarantee it will make a difference. how much you or your other gear (cables, room, speaker dB thresh-holds, etc.) play as much of a part in being able to reproduce the differences is a whole different topic.
another example - stand outside in your underwear in the sun. then do the same thing with a blanket around you. tell me there is not a difference related to the time that you stand there?
cause damage to the tube? not likely. will it affect the tone? you need to remember tubes are a vaccum - voidless of all air as long as the barium is a mercury colour. to say it does not make a difference is to say the cloth of your guitar cabinet does not make a difference. just because you cannot hear a difference does not mean others can, have, or will.
in this topic the literal meaning of this acronym is to be mentioned:
YMMV.