The Andorrans nowadays don’t stamp any passports unless you ask for it, and their stamps are not Schengen stamps, nor do they issue exit stamps
The Faroe Islands, although not itself part of the Schengen Area, is a territory of a Schengen country and has an open border with the Area, so unless you fly to/from Edinburgh, there are no immigration checks. In any case, when there are checks, Danish Schengen stamps are used.
An Andorran entry stamp
And a Faroese exit stamp
I asked two friends who, separately, made the trip from Barcelona Spain to Andorra, and back to Spain. One was traveling on a Canadian passport, the other Turkish. Neither one received a stamp at the Andorra border—but they did not explicitly ask for one either.
I am not sure why your ask Will I get a Schengen entry stamp when leaving Andorra or Faroe Islands? because as I understand your plans it seem what you need is an exit stamp from Schengen, or an entry stamp for somewhere outside the Schengen agreement.
Obtaining such a stamp from Andorra would seem to be easy. From Wikitravel:
A souvenir passport stamp may be available at the border on request.
and
Border control officers at both sides are generally fine.
Taken together I interpret "may" to mean something like "all you have to do is ask" – rather than, say "may … or may not".
At least you would appear to have four chances: out/in Schengen and in/out Andorra with any one of those possibly sufficient.
There is some advice about The Faroes here:
you won’t get a stamp if you arrive from Denmark (you could probably ask for one at arrival, but that would be atypical I guess)
you can also ask for a stamp at the airport or the very least at the police station in Tórshavn
When I left though, there was an immigration guy who checked passports/ lengths of stay (from entrance into Iceland I guess??). He then stamped my passport with an exit stamp that said DK — Føroyar.
I’ve heard they check all documents upon entering by ferry
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