Push-ups alone are not a direct solution for weight loss or fat burning.
While they can help build muscle and improve overall fitness, weight loss depends on creating a calorie deficit through a balanced diet and increased physical exercise.
Push-ups can help with the latter. Alongside other resistance exercises, push-ups will help strengthen your muscles and ensure the majority of weight loss stems from fat stores not muscle stores. This is valuable for a number of reasons not least of which is better body composition and protection against rebound weight-gain.
However, for effective weight loss, it's important to combine resistance exercise (like push-ups) into a more comprehensive fitness plan that includes plenty of cardio exercise (such as walking) and a healthy eating plan which is easily controlled for calories.