Dieting should not slow down your metabolism. But if you trick your body into thinking you're starving itself, it will.
Here are some tips to help keep your metabolism when you're trying to eat healthier. First, don't skip meals. Especially breakfast. If you don't eat breakfast and you don't eat until lunch time, your body is going to think that you are in starvation mode, so therefore, whenever you do decide to eat, your body is gonna hold on to every single calorie that you put in, because it thinks that you're starving.
The next this is, choose foods that are high in protein, and low in carbohydrates. The reason for this is that if you eat food high in carbohydrates your body is gonna release more insulin and this is going to lead to greater fat storage. Which is going to actually slow down the amount of calories that you're able to burn.
Lastly, you want to exercise. Make sure to try to take a few minutes to get your heart rate up throughout the day. Ideally you can exercise for 30 minutes, 3 to 4 times per week, but if you can't fit that into your schedule do smaller things like take the stairs, park further away.
If you do these tips while your trying to eat healthy and diet, you should not slow down your metabolism. The other thing is, you should try to eat foods high in protein. And when you do exercise you should also try to build some muscle mass. This way, throughout the day your body will be rebuilding its muscle and this helps to keep your metabolism up, by burning calories throughout the day.
I hope this answered your questions about dieting and metabolism.