The following is a quote from http://www.dinonline.org/2010/05/30/cooking-in-the-garbage-bin regarding using the same garbage bin for both meat and milk, which I think is directly relevant to your question:
In answering this question, it is
important to note that although
various authorities quote the concept
of a davar gush (solid body) as being
able to “cook” (see Magen Avraham 318,
Nishmas Adam 20:3), it remains a
chumra that is widely disputed (see
Rema, Darchei Moshe 105:4, Pleiti
(ibid.), Chasam Sofer YD 95, Shut Pnei
Yehoshua 2:23, and others). This
opening point gives the following
considerations significant weight.
Davar She’eino Miskaven: See Iggros Moshe YD 39, Yad Yehudah 92, among
others, who write that (like other
prohibitions) the prohibition of
cooking meat and milk does not apply
where it is a davar she’eino miskaven.
In our case it is hard to call the
single garbage bin a “pesik reishei”
for mixing meat and milk — by the time
the teabag gets thrown in it might be
cold, it might not come into contact
with the meat, and so on.
According to many sources, the prohibition of cooking meat and milk
together is due to the concern that
one might come to eat the mixture (see
Kesef Mishnah Tumas Mes 1:4).
Accordingly, where there is no chance
of this happening, some authorities
are lenient (Dovev Meisharim, Har
Tzvi).
Even if a davar gush cooks, it might not be called “derech bishul” to
prohibit the cooking of meat and milk
together (see Pri Megadim, Intro.,
s.v. Od Adaber).
Because both foods are (usually) already cooked, some authorities
maintain there is no longer a
prohibition of cooking meat and milk
(see Chasam Sofer YD 82; Pri Toar
87:3; Aruch Hashulchan 87:13; Machazik
Berachah 87:12).
The combination of the factors above
would seem sufficient to permit the
use of a single garbage bin for meat
and milk.
