I found in a beis medrash what appears to be a kosher mezuzah. I posted signs in the beis medrash and in the lost-and-found sections of the local newspapers, but the owner has not contacted me.

Q: What should I do with the mezuzah?

A: When one finds a lost object with an identifying mark (siman) and it is possible that the owner is not aware that it was lost, the finder must return it to the owner. Even if a long time passes so that it is clear that the owner realizes his object was lost and despaired of recovering that object, nevertheless, the finder may not keep that object for himself. Since it is possible that it was found before the owner despaired (yei’ush), the finder became a custodian of the object for the owner. The owner’s subsequent despair does not relieve the finder of his responsibility to return the object to the owner (C.M. 262:3). The halachah that applies in such a case is that the finder must keep it until Eliyahu Hanavi comes and identifies the owner (C.M. 267:15), and during that time the finder is the custodian.

In your circumstance there are two options:

1) Halachah permits you to acquire the lost object for yourself. When one finds a pair of tefillin, he may appraise their value and use them immediately (C.M. 267:21). The rationale is that tefillin are readily available and the owner will not mind if someone takes his lost tefillin and repays him with money since he can easily purchase another pair and is happy that someone will use his possession to fulfill a mitzvah (see Sema 30; and Shach 16 and 72:8).

2) In your circumstance it is permitted to use the mezuzah even as the custodian. Generally a custodian is not permitted to use the object in his care and one who does so is a thief (sho’el shelo midaas, gazlan), and possibly a sholei’ach yad (the term for a custodian who uses the object placed in his care for personal use [C. M. 292:1]). However, in this circumstance what is the difference between storing it in a cabinet or drawer or placing it in a case and affixing it to a doorway? Therefore, since it is not necessary to own the mezuzah to fulfill the mitzvah, the custodian would fulfill his obligation (Har Tzvi, Y.D. 238; Minchas Chinuch 423; cf. Daas Kedoshim 289:5). Furthermore, there is a general allowance to use another person’s object to fulfill a mitzvah when there is no concern that it will become damaged since we assume that people are happy for their objects to be used for a mitzvah (O.C. 14:4).

However, if you do not acquire the mezuzah for yourself it is prohibited to use it in a manner in which it could become damaged or stolen — for example, by using it on an outside doorway.