A plate armour can be noisy:
Note that the main source of noise in these armours is the tassets hitting the thigh armour when moving. If a partial armour without leg armour was being worn, it would be much quieter. Note also that tassets were often lined, which would greatly reduce the noise (Oakeshott 2012, pp 206-207). Other pieces of armour were also often lined. At least one function of lining armour was to reduce noise.
Mail is fairly quiet. An unlined coif moving over a mail shirt can produce some noise, and an unlined skirt can make noise on mail leggings (or plate thigh armour). As with plate armours, linings or clothing worn under the armour to stop overlapping pieces making direct contact will reduce noise. It's much harder to find good examples of the noise from mail compared to the plate armours above; here is an audio-only example:
Note that the noise from the mail is quieter than the footsteps.
For Japanese armour noise,
Thee armours have many overlapping pieces, and tassets that overlap metal thigh armour (which are one of the bigger sources of the noice), but not much noise is produced.
So, in summary, armour produces some noise, but it should not produce a lot of noise. If unlined plate pieces (or mail) move a lot and can hit a piece of plate they overlap, they can make a lot of noise (as in the 2nd example give). Tassets on thigh armour are the main offender. However, this was usually stopped by lining the armour (e.g., lining the tassets).
Reference:
Ewart Oakeshott, European Weapons and Armour: From the Renaissance to the Industrial Revolution, Boydell Press, 2012 (first published Lutterworth, 1980, reissued Boydell, 2000 and 2012).