The liver is rather unique in being able to regrow after partial donation. More interestingly, growing organs in general uses loads of energy, and while I'm not exactly advocating for advertising liver donation as a weight loss strategy, how many calories exactly does it take to regrow a liver?
To get the answers started, though admittedly with a very rough estimation, I'll add my own idea of a lower bound of calories burned to regrow an entire liver.
Because basic laws of thermodynamics state that to produce 100 calories worth of food requires at the very least 100 calories worth of energy, an estimate on the nutritional content of the liver itself should be a reasonable lower bound for how many calories it takes to regrow a liver.
According to Nutritiondata.com, raw pig liver has 134 calories per hundred grams, and raw cow liver has 135. Given the relative consistency, right around 134 should be a reasonable estimate for the caloric content of human liver, not that any of us really want to test that one out. Now according to Hypertextbook referencing a number of sources, average mass of a human liver is around 1.5 kg.
Finally, UCSF Medical Center states that the majority of a liver is regrown in the first two weeks. In theory, this would mean that in the case of a whole liver donation, half the liver would be regrown in the first two weeks after donation. However, whole liver donations tend to not lead to regrowth, with this study (unfortunately paywalled) stating that around 25% of the liver needs to be left to initiate regrowth. For arguments' sake however, and considering we're already vastly underestimating the caloric needs for liver regeneration by assuming 100% efficiency, let's roll with it.
So 1.5 kg of liver at 134 calories per 100g yields 2,010 calories per liver, of which 1,005 would regrow in the first two weeks, or no less than 71.79 calories per day to regrow a liver for the first two weeks after loss/donation. Notably, Raymond Lindeman's 10% law implies that regrowing a liver may burn 10x this number of calories or more, so I'm really curious to see what answer someone who knows more than I do might provide.