Should you be eating back the calories you burn?

Do you mean calories burnt during exercise?
If yes then I wouldn't eat them all back I'd eat half of them back or if you haven'g burnt much don't eat any back unless you're hungry. As you can't be 100% sure how many calories you've burnt eating them all back may end up meaning you've eating too many and gone over your calories.

MFP gives you a deficit for weight loss that doesn't include any exercise. This is so that you lose weight WITHOUT working out. If you workout your daily calorie burn increases, so your calorie intake needs to go up to maintain your deficit. Having too large of a deficit can cause you to be really tired, hangry, lean body mass loss, and there is even the possibility of it leading to a binge.

Example:
MFP estimates I burn 1860 at the Lightly Active setting in a 24 hr period without exercise.
To lose 1 lb per week:
1860-500= 1360
1360 is my NET goal. I should eat 1360 on days I don't workout.

If I workout and say burn 500 calories for the day from exercise, then:
1860+500 = 2360
I burned approximately 2360 for the day.
I could just eat the 1360 and shoot for 2 lbs per week loss. However, at my weight that would mean losing lean body mass as well as fat. The max I should aim for if I want it to be mostly fat loss is 1.2 lbs per week or 0.5 lbs if I want it to be easier to stick to.
To lose 1 lb per week with my exercise, I should eat 1860. This keeps my deficit from getting too large, fuels my workouts, and doesn't leave me feeling run down.

It is recommended to start with only eating 50% back and then adjust based on your average loss of a 4-6 wk period. Losing faster than 1% bodyweight, more than 2 lbs per week, or just faster than you are aiming for; eat more of those calories. Losing slower than you want (providing that want isn't more than 2 lbs per week); eat less of those calories and double check your logging for errors (there are a lot of very wrong database entries and if using measuring cups you might want to consider a food scale for better accuracy with portion sizes).