Your question is effected by many halachic concepts, including some that are too complex to discuss thoroughly in this context, but we will refer to as much as needed to give you a ruling for your case.
It is forbidden to kill animal life on Shabbat, and this prohibition has elements of severity that exceed the prohibition to trap them.[1] However, there are several potential reasons for leniency in your case. We are assuming that your main intention in closing the closet door is unrelated to killing moths.[2] There is sufficient time to kill the moths, larvae, and/or eggs during the week.
If it is certain that closing a door will kill moths, this unintentional but definite and otherwise desired result is called a p’sik reishei, a violation of Shabbat that is usually forbidden on the level of Torah law.[3] On the other hand, killing an animal through poison is not a classic direct action; one puts out poison, which later kills without human involvement. There is a machloket Acharonim regarding whether one is allowed to put poison in front of animal on Shabbat so that it will eat it and die. The Shvut Yaakov[4] makes a strong argument that this is a situation of gerama (indirect causation), which can be permitted when certain other alleviating factors apply. Here, the killing is a melacha she’eina tzricha l’gufa (it is not done to make use of the final result, i.e., the dead moth), which almost all Rishonim agree is only a Rabbinic prohibition. The Shvut Ya’akov says that with the combination of these mitigating factors, placing the poison can be permitted to avoid tza’ar. Presumably, it would also be permitted in order to avoid a loss). Even in the case of a p’sik reishei of a direct but unintentional action, significant opinions permit the action if only a Rabbinic prohibition is involved.[5]
Your case, however, is different in both the direction of stringency and leniency. Here, the moths are not enticed to eat the poison, but rather the fumes reach the moths, which is more direct. (Rav Ovadia Yosef[6] makes this distinction regarding a case of spraying insects with poison.) On the other hand, here the substance is placed before Shabbat, and you are inquiring about closing a door on Shabbat. That action does not create the fumes or direct them at the moths; it simply stops the fumes from dissipating. One could argue that this should still be forbidden. After all, in a parallel case, when one closes a pot of not yet cooked food so that the heat will not escape and the food will thereby cook faster, most consider the action a full violation.[7] However, it is unclear that this model should be applied in all contexts. (The matter demands further investigation beyond our present scope).
The strongest reason to permit closing the closet door is the uncertainty whether any moths will die on Shabbat as a result of that action. First, it is not certain that there are any moths or larvae there at this time, especially given that ongoing use of the substance has already eliminated or at least decreased their population. This may make the action a davar she’eino mitkaven, performing an intrinsically permitted action when there is only a possibility – not a certainty – that it will cause an unintentional forbidden result, which is permitted. There is major discussion regarding whether this leniency applies even when the uncertainty has to do with an existing situation (i.e., are there moths?).[8] (It is possible to counter that in a case in which if there are moths, they will certainly die immediately, it should be forbidden to enter the possibility of a violation through p’sik reishei.[9]) Furthermore, our research suggests that it is unclear that moths that are in the closet will definitely die on Shabbat because the extermination process is a slow one. Therefore, it should be permitted to close the door assuming that one is not doing so with the intention to kill the moths as quickly as possible.
In summary, because of a few possible factors and perhaps their confluence, it is permitted to close the door.