Hey Hapi', I went back and dug up this info from RME on dither, relative noise and the effects of bit reduction. It's pretty old (from 99') but still relevant. I found it a good primer for laying this stuff out in a clear way.
http://www.rme-audio.de/english/techinfo/dither.htm
From 'Dither in the ADI-1 / ADI-8 PRO 'Remarks about the Need for Dither'
'.. However, their purpose is often to prove that A/D converters without dither would produce horrendous distortion, and therefore sound terrible. This is actually (willful?) deceit, as truncation of a digital test signal has little or nothing in common with truncation of an A/D converter's signal, and certainly doesn't lead to such 'catastrophic' results as seen in fig. 2. This is mainly thanks to inherent noise in the converter, as can be seen in the following measurements using an ADI-1. '
Thanks, mixsit. So if I understand this correctly, you're saying that certain low-level noises that would be inaudible in 24 bit resolution become audible (or more likely to be audible) when the tracks are converted to 16 bits. Have I got that right?
Yes. There was a nasty thread a while back here at HomeRecr's where fur was flying as to whether +/- dither could be proved to be heard at all. (ducks for safety..
If so, and if it turns out that my workstation really doesn't have a built-in dither option, could I perhaps solve the problem by applying a noise gate to all my tracks during mixdown?
No gate. (See the RME thing

a) Whatever noise (this one of the base quality' or lack of you have in your recordings) is also happening while the music is playing.
b)This is way down, likely below the noise in and from your recorded signal.
Also, isn't it likely that a lack of dithering might not even be noticeable as long as I stick to recording pop and rock tracks and avoid quiet string ensembles?
That's may take on it.
BTW, which Yamaha ya got?
ps. A nice experiment-Try recording in 16 bit (let alone 24) something nice and open' like accoustic guitar low enough to hear the converter noise above your analog noise. Now consider the dynamic range you have above that.