Does Dorico behave differently if there is no key signature, compared to setting C major?
Assume tonal/modal pre-1750 music with no transposing instruments! 
From the manual:
“In an open key signature, the spelling of accidentals is based on the current direction of the music. If the music is rising, sharps are preferred, whereas if the music is falling, flats are preferred. There is no hierarchy of pitches in an open key signature, so the same pitch might be spelled differently each time it appears depending on its context, even within a few bars.
In a C major or A minor key signature, accidentals are spelled based on the context of the major or minor tonality implied. For example, in C major, sharps in general are preferred, whether the music is going up or going down. Similarly, in A minor, G♯ in particular is preferred, whether the music is going up or going down, as G♯ is the leading note in A minor.“
(Polite hint to RTFM 
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But will it alter existing accidentals if they key sig is removed?
No. But the C major key signature is an insurance policy - you might as well hang on to it in case the software changes in future…
I mean, what does it cost you to throw in a C major key signature? 5 seconds?
Easy, Tiger…r 
I’ve got some imported XML passages that have come in with no key sig, so yeah I’ll probably assign C. Or even A minor if I feel like it.
Fair enough. My reading of the documentation is that it’s advisable to use a key signature if you’re working within a key, even if that’s C major or A minor. I can’t think of any obvious differences to the way that Dorico displays the music, but there may be things going on under the hood that we don’t know about.
(Oh, and Leo’s a lion, not a tiger 
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Harmonically speaking, I’m in G minor/D major, but it’s all done with accidentals.