It is highly unlikely that a food can possess negative calories (going by the below definition of negative calorie), and even less likely that it per se would induce weight loss. Negative calorie foods tend to be satiating though, so their consumption could lead to weight loss through suppressing food intake.
The concept of 'negative calorie' food is a food that itself contains a very small caloric value (usually under 10kcal per serving) that takes more energy to digest than it provides. Theoretically, the end result would be a food which induces a caloric deficit after consumption rather than a surplus.
Not much evidence exists on the topic of 'negative calorie' foods in human or animal consumption. Thus the following is part conjecture.
It is highly unlikely that a food can have negative calories. The concept of 'the thermic effect of food' (how much energy is required to digest a food) is greatly tied in to the caloric content, with more energy being required to digest more calories. Regardless of how little calories a food contributes, if the Thermic Effect of Food fluctuates in the 3-30% range it cannot exceed 100% by itself.
It is possible there could be a bioactive compound in the food that enhances metabolic rate and thus exceeds the caloric value. Some compounds are found in foods and able to stimulate the metabolic rate either in intervention studies (Synephrine and naringenin from grapefruit) or in theory (Higenamine from Nandina fruit); however, this is going to depend on the food ingested rather than being applied as a blanket statement to low calorie foods.
Even if we are to assume bioactives that can enhance the metabolic rate, a metabolic rate that is increased by 10kcal is not likely to result in weight loss in any short timeframe if at all.
The possible benefit of 'negative calorie' foods is that they also tend to have high water contents (celery, tomatoes and lettuce are commonly cited) and thus may offer more satiety, which may manifest itself through less food consumption overall. This lesser food consumption would lead to weight loss.