I'm not sure why it happens, but I shook terribly while I was on prednisone. I didn't notice it getting worse when it's warm - mine got worse when I was nervous. I know that certain kinds of tremor (whether a med side effect or not) can be worse when you're trying to perform an action that requires steadiness - people might not shake at all when resting, but a movement such as holding a cup or trying to write brings on or worsens the tremor, which is possibly why you notice it more when eating and using cutlery.
For me it also started before tapering, so it might not be the tapering that brought it on for you so much as the length of time you've been on pred for. The shaking stopped for me pretty much as soon as I was off it though, so hopefully you won't have to deal with it too much longer.
