The fundamental principle permitting pre-Shabbat initiation of processes that continue automatically is established already in the Mishnah (Shabbat 17:2), where Beit Hillel permits soaking ink and dyes before Shabbat even though the process continues into Shabbat, and in the Gemara (Shabbat 18a), which adds that one may load an olive press or wine press before Shabbat so that production continues on Shabbat on its own.
The Rambam (Mishneh Torah, Sabbath 3:1–3) codifies this principle explicitly, stating that it is permitted to begin a labor before Shabbat even if it is completed on its own on Shabbat, since the Torah's prohibition applies only to performing labor on Shabbat itself, and benefit from what is produced automatically on Shabbat is permitted.
The Mishnah Berurah (252:1) confirms that even where the action completed automatically would constitute a Torah-level prohibition if done on Shabbat itself — such as kneading or dyeing — beginning the process before Shabbat is permitted, since the forbidden act is done at a permitted time.
The Shulchan Arukh (Orach Chayim 338:3) rules that one may set a clock-bell mechanism before Shabbat so that it will ring throughout Shabbat, establishing a precedent that configuring an automatic mechanical device before Shabbat to operate during Shabbat is permitted.
Peninei Halakhah (Shabbat 17:26) states explicitly that one may install a Shabbat timer on an outlet and connect to it an electric oven or fan, setting the timer before Shabbat to turn the appliance on and off at desired times during Shabbat, since the prohibitions of Shabbat take effect only from the onset of Shabbat whereas actions performed before Shabbat whose effects continue into Shabbat are permitted.
Rav Moshe Feinstein (Iggerot Moshe, Orach Chayim 4:60) rules that it is strictly forbidden to use a timer set before Shabbat to perform creative labors such as cooking on Shabbat, arguing that permitting this would lead to severe desecration of Shabbat since people would run entire factories and perform all manner of labor automatically.
The Gemara (Shabbat 120b) establishes that the Torah prohibition states 'you shall not perform any labor,' implying that direct performance is what is forbidden while indirect causation is technically permitted, though the Gemara notes that where a person is agitated about his property one may not rely on this leniency lest he come to act directly.
The Shulchan Arukh HaRav (Orach Chayim 252:17) notes that the Torah prohibits only מלאכת מחשבת — intentional, deliberate labor — and that a Torah-level violation requires that the person consciously intend to perform the act, which bears on the analysis of automatically executed processes.
The Talmudic discussion in the Gemara (Shabbat 19) records a machlokes between Rabbi Yishmael and Rabbi Akiva over whether one may complete before Shabbat a process whose continuation on Shabbat occurs on its own, with Rabbi Yishmael permitting and Rabbi Akiva forbidding, reflecting the underlying tension in permitting automatic Shabbat processes.