There are a couple possible interpretations of the question.
One is to take a song, let's say a simple Am - Em - Am progression, and to tune your guitar down a half step, and play Abm - Ebm - Abm.
There is some difference in sound just from being in a lower key. When you go lower, chords gradually sound "muddier". Is this considered a timbre change from the tuning? Well, when a metal guitarist talks about "how much beefier drop B sounds", this is probably what they mean.
Well, what if we were to tune our guitars differently just for the timbre and keep our music in the same (concert pitch) key?
There are two major contributing factors here:
A longer guitar string gives a brighter tone, whereas a shorter guitar string gives a darker tone.
A thinner guitar string gives a brighter tone, whereas a thicker guitar string gives a darker tone.
(You might debate my choice to use the words "brighter and darker", but there is definitely a particular type of sound that comes from longer, thinner strings*.)
Given that, let's consider a tune that doesn't use open strings, which we could easily play in different tunings. Let's imagine playing:
Let's tune down a half step now. If we want our strings to have the same tension as our E standard setup, we change our strings to a thicker set. Also, since we want to play in the same key, we have to move all our chords (/dyads) up a fret. Now every note we hear is coming from a thicker, shorter string now, so our Eb tuned guitar sounds darker.
But wait! We could also move the entire riff up a string! Now everything is being played on a thinner, longer string, so it sounds brighter instead! Which is it?
By playing every note in the lowest position possible, we can see that our low E, F, F#, and G got slightly darker, our G# is significantly brighter, our A, A#, B, and C got slightly darker, our C# is significantly brighter, and so on...
We could always choose to play notes higher up on the neck for a darker tone, so these statements describe how the brightest version of each note changed.
So what's the net timbral effect of retuning? It depends on what you're playing, and how you're playing it.
*Jason Pelc - Solving the Sound of a Guitar String | http://large.stanford.edu/courses/2007/ph210/pelc2/
I will update this with a better source if I ever do find one. This gives a model of a stiff string oscillator (like a guitar string) where high frequency harmonics are damped more on strings with a higher ratio of width to length- in other words, shorter and thicker strings are darker. There are also synthesized demonstrations of different strings, which will give you an idea of what I mean by "brighter or darker".