The short version is that it might have an effect, but there is no evidence to date.
I was surprised to find that there are actually quite a few scientific studies on this subject. There's an even larger literature on the effects in animals than in humans.
A good survey paper on the area (which may be paywalled) is found here. The crux of it is that the best designed studies to date do not find any significant difference in fatty mass changes between those who take quite large doses of CLA (roughly equal to eating 250ml of ghee a day, if my understanding of ghee's CLA composition is correct), and those who just eat an olive oil capsule instead. This holds across many populations. Experienced weightlifters, novice weightlifters, and overweight or obese persons. The only human studies reporting significant effects tracked changes over a quite short period of time, or else had systematic problems.
An example of a study that has some problems can be found here.
The study above shows a decrease in fatty mass for overweight and obese individuals who consumed concentrated CLA (roughly equal to the amount found in about a half a litre of Ghee, from what I can ascertain) in daily doses, versus those who consumed an equal amount of olive oil (the control).
The improvement is statistically significant, but extremely small. Over 12 weeks, the average subject on olive oil gained about 1.5kg (~3lb) of fat, while those taking CLA lost a small amount of fat instead, but also gained about 1.5kg of lean mass.
Even if the study did control properly for exercise, they found no benefit beyond a dose of about 3.5 grams/day of CLA. A weighloss supplement that cuts ~1/4 pound per week from an overweight person is not particularly impressive. The person could just as easily drink one less glass of soda (or milk!) per day to lose at the same rate.
However, the study included sending an offer of training from a local gym to all participants. Those in the higher CLA dosage groups were found to have engaged in an average of 1 more hour of "heavy training" per week by the end of the study than at the start. Over a 12 week period, that could easily be enough to result in a recomposition of just 2kg or so (i.e. trading 2kg of fat for muscle), especially in novices engaged in something like resistance training. The changes in BMI between the groups were not significant at the end of the study, suggesting that all participants gained the same weight, just some of them put it on as muscle.
So, to sum up, there do not seem to be any reputable studies showing a benefit in body fat loss from consuming CLA. There seems to be a lot of interest in it because it is present in certain health foods (like Ghee) though.
As an aside, I'm personally very sceptical of claims that any food (especially what is essentially liquefied fat!) will magically cause weightloss or fat loss. Losing weight isn't complicated. To get rid of a few pounds of "stubborn" fat, eat fewer calories than you burn (your TDEE) and exercise more for a few weeks. As a general rule, all the rest of it is noise or tiny effects not worth worrying about. Remember: if there was a miracle cure for weightloss, everyone would be thin already. Companies would be lining up to sell it.
