Gemara

40b (Beraisa - R. Yehudah): A woman may leave a flask of oil in front of a fire, not to get cooked, rather, only to get warm.

51b (Beraisa): One may not crush hail or snow in order to melt it. One may put it in a cup or bowl.

Rishonim

Rambam (Hilchos Shabbos 21:13): If one squeezed pickled or cooked vegetables, if he did so to soften them, it is permitted. If he did so to extract the liquid inside, it is forbidden. One may not crush snow in order that [it will melt, and] the water will flow. One may crush it into a bowl or cup.

Magid Mishneh: The Ramban and Rashba say that a Tosefta (4:15) permits crushing into a bowl. The Rashba forbids when he intends for the water to flow, due to a decree lest one squeeze Peros destined for liquids. It seems that the Rambam agrees, for he brought this law with laws of squeezing. We learn from them that there is no Isur of Nolad. Therefore, one may put a pot or a food with congealed fat in the sun or near a fire, and there is no problem of Nolad. Also the Ramban said so.

Ran (23b DH Aval): A Tosefta connotes that one may overtly crush the snow or hail into the cup, for it says "but he may crush into a bowl." This connotes that the concern is due to resemblance to Melachah. Alternatively, it is a decree lest he squeeze Peros destined to [be squeezed into] liquids. It seems that the Rambam holds like this. This is only if he overtly squeezes. If he put them in the sun and they melted, or even near a fire, they are permitted. This is not Nolad. This is unlike liquids that flowed, for these are considered liquids in every way even when they are solid. However, Sefer ha'Terumah forbids due to Nolad [and Muktzeh]. He forbids putting near a fire a pot in which the fat congealed, for initially it is thick and solid, and now it melts and becomes clear. Based on this, even in the sun the [melted] fat is forbidden due to Nolad. The Ramban asked that fat is always a food, so one may crush and squeeze it. L'Chatchilah one may squeeze Peros that are not normally squeezed (144b). Even juice that came out by itself from strawberries and pomegranates, if they are destined to be eaten, is permitted. Therefore, whatever exudes by itself, whether in the sun or opposite a fire, is permitted. I say that this is not difficult. Fat, as long as it is not congealed, is clear and flows. This is its normal state. Even when it is congealed, it is like Peros destined for liquids.

Rashi (51b DH Kedei): One may not crush snow or hail in order that [they will melt and] the water will flow, for this is Molid. This is like a Melachah of creating the water.

Rashi (DH Aval): One may put them into a cup of wine in summer to cool it, even though they melt by themselves, and he is not concerned.

Poskim

Shulchan Aruch (OC 320:9): One may not crush snow or hail, i.e. to break them into small pieces in order that the water will flow.

Beis Yosef (DH Ein): Sefer ha'Terumah forbids what flows, even if he left them in the sun and they melted or flowed by themselves, due to Nolad. One may put them in a bowl or cup because they mix with the wine or water, and they are not recognized. I wrote in Siman 318 that most Poskim argue with Sefer ha'Terumah. It seems that the Isur to crush is only when he cuts into small pieces, like Rashi says. One may break a piece from it. The Tur connotes like this. He permits breaking ice to get water underneath. Even though small pieces are strewn when he breaks it, we forbid only when he crushes in order that the water will flow. Water does not flow due to these small pieces. Even if a little flows, it is not significant. Also, he does not intend for it. Also, the water goes to waste. The custom is to permit. I wrote this because I saw one who disputed this.

Kaf ha'Chayim (59): The Mechaber and some of the Acharonim forbid crushing into a cup, so we follow them.

Magen Avraham (14): Based on what the Rema wrote in 318:16 (brought below), also here it is forbidden due to Nolad, unless it is mixed with water.

Gra (DH v'Chen): The Rashba forbids only lest one squeeze. Therefore, only overt crushing is forbidden. A proof is from 40b. Stam oil is congealed; see 318:16 (below).

Note: The Mordechai (247) says that Stam olive oil is congealed. The Ritva (51b DH ul'Inyan) says that Stam oil in the days of Chazal was congealed. Zevachim 91b discusses taking a Kometz of [olive] oil, i.e. curling three fingers around it and wiping off the excess that sticks out on each side. A liquid would flow out of his fingers!

Mishnah Berurah (33): This is like Melachah. He creates water. It is forbidden mid'Rabanan. Alternatively, it is a decree lest he squeeze Peros destined for liquids.

Shulchan Aruch (ibid.): However, one may put it into a cup of wine or water, and it melts by itself, and he is not concerned

Bach (DH u'Mah she'Chasav Af): Here the Tur permits because he does not overtly crush. In Siman 318, he brings Sefer ha'Terumah, who forbids putting a meat pie near a fire, for the fat will melt, even though this is not overt! I answer that there it is Nolad because the fat is recognizable. Here, the melting snow or hail is not recognized in the cup. It mixes and becomes Batel.

Magen Avraham (13): The Shulchan Aruch connotes that one may not overtly crush. The Ramban, Ran, Rambam and Magid Mishneh bring from the Tosefta that one may crush into a cup. The Isur is lest one squeeze Peros destined for liquids. Since [the melted water] mixes with water [in the cup], Chachamim did not decree. See Sa'if 11. (The Shulchan Aruch says not to wash hands if there is snow or hail in the water, lest he crush it. If one does, he must be careful not to crush.)

Mishnah Berurah (35): One may benefit from them. Also, l'Chatchilah one may put it there, for it melts by itself. Likewise, one may put in a cup. According to the Rema in 318, also here we should be stringent. Even though it melts by itself, it is forbidden due to Nolad. This is unlike in a cup (with water), for there it is mixed with the water and it is not recognized.

Kaf ha'Chayim (63): There, the primary opinion permits, just l'Chatchilah we are stringent. The same applies here. Therefore, one should add water before putting it in the sun or near the fire. If he has no water and needs to drink, he may be lenient (to put it there without water).

Kaf ha'Chayim (60): Some people put water on yogurt or Leben (congealed milk) and dissolve it with a spoon to make it like milk again. Ben Ish Chai equates this to crushing snow into a cup. One should be stringent, but in a place where the custom is to be lenient, we do not protest, for they have on whom to rely. One who is stringent will be blessed. One may be more lenient about dissolving sugar into water. For the sake of guests, one should not be stringent. All permit to shake the cup without stirring.

Tzitz Eliezer (6:34:15): We permit putting fine sugar in water or tea, and stirring to dissolve it quicker.

Tzitz Eliezer (4): For the sake of a Mitzvah, we permit a Pesik Reishei that one does not want. (E.g. one puts ice in an empty bowl, so he will have ice to put in his drink, and some of it will melt in the bowl.) Oneg Shabbos is considered a Mitzvah.

Tzitz Eliezer (9): If something freezes when it is put in a refrigerator, this is considered automatically, so it is not Molid. Even according to the Rema, we permit for the need of guests or Oneg Shabbos.

Tzitz Eliezer (19): People remove food and drink from the freezer, and they are not careful to eat them before they thaw out. According to Sefer ha'Terumah and those who hold like him, they are forbidden due to Nolad once they thaw out.

Shulchan Aruch (ibid.): Similarly, if he put them in the sun or near the fire and they melted, they are permitted.

Kaf ha'Chayim (62): This is permitted l'Chatchilah (R. Akiva Eiger and Rabbeinu Zalman).

Ashrei ha'Ish (2:30:7, citing ha'Gaon R. Y.S. Elyashiv Ztz"l): One may not take ice out of the freezer if it will melt there. This is considered an action. One may pour water onto a tray in which ice cubes formed to remove them, even though he overtly melts, since only a little melts. One may stir a cup with water and ice cubes, even though it causes them to melt sooner. If water melted from ice itself, one may not drink it if he has other water. If a fruit became wet from such water, he must rinse it off.

Shulchan Aruch (318:6): One may put a meat pie near a fire where it is Yad Soledes Bo, even though the congealed fat will melt.

Rema: All the more so a pot with congealed gravy is permitted, for when the fat melts it is not by itself. Some are stringent. The custom is to be stringent. However, when necessary one may rely on the first opinion.