If someone left soup on a hotplate when it was off, forgetting that it was on a timer and would come back on later, can he eat it after the soup is reheated, because you can't do this on Shabbos?
There is a machlokes as to whether you can purposely do this or not. Most poskim say you can't but R' Ovadya Yosef allows it. See here
While one mustn’t place a davar lach, a boiled food with liquid on the stove or hotplate on Shabbos, R’ Ovadia Yosef (Yabia Omer 10 OC:26) allows one to place such a pot on the hotplate while it’s off even though it will later switch on through a timer.
Most poskim, however, disagree. R’ Tzvi Pesach Frank (Har Tzvi OC 136) compares using a timer to later heat food to placing food on a stove that will be lit soon, which he argues is assur deoraisa. R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 1:26), the Chazon Ish (38:2-3) and R’ Benzion Abba Shaul (Ohr Letzion 2:30:18) write that while it isn’t assur deoraisa, one still can’t do so on Shabbos because it is a problem of gerama, causing something to happen (though one could ask a non-Jew to do so while it was off).
Either way, if you do this, you are still allowed to eat it
.. The Gemara (Kesubos 34a, Chullin 15a) writes that if one transgressed a melacha on Shabbos, there is a machlokes as to whether they or others can benefit from it on Shabbos or afterwards. The Gemara discusses whether this prohibition is mideoraisa or miderabanan. Rambam (Shabbos 6:23) and the Shulchan Aruch (OC 318:1) follow R’ Yehuda and write that if one accidentally cooked food on Shabbos, everyone must wait until after Shabbos to eat it. The Vilna Gaon (Biur Hagra, OC 318:1), however, follows Tosafos and others who pasken like R’ Meir, who holds that may one eat such food on Shabbos.
The Mishna Berura (318:7) writes that while we should generally follow the Shulchan Aruch on this, if necessary, one can rely on the Vilna Gaon. Thus, if this was one’s main dish for their Shabbos meal, one could still serve it.
The Pri Megadim (Eshel Avraham 318:10) and Mishna Berura (318:2) write that when there is a machlokes as to whether something is an issur or not, one doesn’t need to wait to benefit from it. Following the opinion in the Gemara that this prohibition against benefitting from forbidden melacha is miderabanan, we apply the rule of safek derabanan lekula, we are lenient in matters of Rabbinic doubt. As there are rishonim who maintain that ‘ein bishul achar bishul’ applies equally to liquids, one who accidentally reheated a liquid would not have to wait to eat it.
In conclusion, one may serve food containing liquid that was accidentally reheated on Shabbos.
Yes, even if he intentionally reheated the soup on an existing fire by mistake, he can eat it afterwards. The Chafetz Chaim in Biur Halacha (318, 4) rules this way since there are Rishonim who permit Bishul Achar Bishul with liquids that were cooled, and therefore we will not forbid its consumption after the fact.