I have found some good information on dovecotes in France in the Middle Ages:
http://members.aol.com/Duiven/dovecote/chapter2/cooke02.htm 
I do have a few questions that aren't answered there.

The time is November 1309, and the place is central France, general
vicinity of Montargis (near Orleans).

- Could there have been an abandoned dovecote that knowing travelers on the
road had taken to using as a rest stop?  (I don't need a backstory on
why it's abandoned.)
- Would it be possible to build a fire inside, say, against one wall? 
I imagine a cylindrical construction with a cupola and four gabled
openings at the top.  Would that be enough venting?  (The first thing
sacrificed for the wayfarer's warmth was the wooden beam-and-ladder
device formerly used for harvesting the squab for the lord's table.)
- Would birds still roost there even if they were no longer being kept
and maintained?  Please say yes if possible.
- How dark would it be inside in daytime?
- What would the floor be like?--anything other than packed earth?
- How thick would the walls be?
- Would there be only an entrance opening, or would there be an actual door?
- In that location, what would be the likeliest material--type of
stone? brick?--and (exterior) color?  This one is not whitewashed.

Because the intended use of this information is for fictional purposes, I don't
need actual historical fact here, only plausibility.  I prefer to take
my liberties with fact wittingly and not out of ignorance.

Thank you,
Archae0pteryx

Thank you very much, Tryx - both for allowing me to post an answer,
and for caring for me.

And I agree with your views on the rough stone walls and the overall
gray appearance of the tower. After all, field stones are available
almost anywhere in the countryside.

And now I'll go to bed, watch the late night movie with Spencer Tracy
and Katherine Hepburn on TV, take two pills against the annoying
pains, and then (hopefully) sleep...

All the best,
Scriptor

Always superlative, even in adversity.  Thank you, Scriptor.

Tryx

Dear Tryx,

This will unfortunately not become an answer since I am ill and, alas,
headaches and fever have temporarily reduced my level of concentration
and my ability to read more than one French sentence. But here are two
or three things I found that may prove useful for you and/or for my
colleagues who will certainly take over at your satisfaction.

I find it absolutely plausible that there is an abandoned dovecote -
or colombier - some travellers know of and that is sometimes used for
shelter. However, if this has done before by other travelers, any
wooden structures (except for the truss) would certainly already have
been dismantled in order to make fire. Besides, I saw some interior
shots of medieval French dovecotes that show the nestholes integrated
in the stone walls, without any wooden construction. But maybe a
nearby dilapidated farmhouse (like the abandoned hut of dovecote
keeper) the can deliver the wood necessary for your travelers.

In medieval France, maintaining dovecotes was a privilege of the
nobility. Certainly the lord who once owned that specific colombier
could have numerous reasons for abandoning it. Or circumstances of any
kind could have forced him to do so. Maybe one year, all the doves
died from some disease, and the dovecote was considered cursed from
that time. Anything is possible. The medieval landscape had abandoned
watchtowers, mills, castles, chapels - an abandoned dovecote seems to
be absolutely realistic, too.

And I believe that there would be enough venting so the travellers
would not suffocate when they make a fire. I have seen images of
dovecotes where the holes for the doves to enter and leave the
building were part of a wooden shutter. With that shutter removed or
already vanished for years, there would be a window that allows enough
ventilation.

I guess there would be birds, even if the dovecote was not maintained
any more. Maybe no doves, but certainly sparrows, tomtits and other
birds that occupy any shelter they can get in winter (I have seen
tomtits in a telephone junction box at the roadside, so why should
they avoid a former dovecote when it gets cold outside?)

Since the door would certainly be closed when your travelers are
inside the dovecote, the window (see above) would be the only source
of light unless they make fire. Considering the pale November sun, I
guess that there would be half-light at best.

The floor could well be made from stone, to keep foxes and wolves from
digging their way from outside under the walls. They do that - my
grandfather told me of foxes that actually made their ways under the
earth into the henhouse, with bloody results.

I think the walls would be about 40 to 50 centimeters thick. Of
course, this depends on what building materials were available, and
how big the dovecote is. This photo of the inside of a rather small
former French dovecote, now converted for other use (the windows are
later additions) give you an impression of how thick the walls could
be:
http://www.castles-france.net/small-castle-sale/images/Dovecote.jpg
The structure dates from a later time (16th century), but I guess that
such basic things were not subjects to radical changes over the
centuries.

There would be an actual door, to keep out predators and - of course -
thieves. This illustration from the "Très riches heures" of the Duke
of Berry shows a dovecote with a door clearly made from solid wood:
http://www.geomaler.de/archiv/krempel/dovecote.htm

May this be useful for you and for the researcher who continues the
work on this question!
All the best,
Scriptor

Oh, dear, Scriptor, I am so sorry that you are ill!  Sounds like you
have some sort of flu.  You must rest, keep warm, and drink lots of
liquids, naturally, and don't be poking about on the Internet trying
to do any work for me.

In reality, though, you have already answered my question with your
comment.  You covered everything but the type and color of
construction material.  I imagine a gray, rough-hewn stone, worked in
an ornamental pattern of some sort on the outside to display the
lord's affluence.  From a distance it would just look like a pale gray
tower.  If this is plausible enough, that's all I need.  I think you
should just post "see comment" and take the credit for this one.

The ladder contraption would be long gone, yes.  That's what I meant. 
It was the first thing burned there by travelers long since.  The hole
in the floor for the shaft has been filled in.

Some fool burned the door, too, which is exactly what the same person
would do today with common resources:  "I'll make myself comfortable
tonight, and this fire will keep wildlife away; and the next guy who
stops by here once the door is gone, well, that's his problem."  So
there's an open doorway where a door used to be, but no more wood.

Thanks, and feel better soon--
Tryx

Yes, indeed, Scriptor, Gute Besserung, and get well quick!
Down here, I would recommend a glass of hot Apfelwein with honey.
Myoarin