Yes, you should try to remove it as soon as possible.
When a bug is smashed into the front of your car all of its guts are allowed to mix together. Most bugs will result in an acidic goop that sticks to your car. You can see it on your windshield, but you won't see it very well on your front grille until the damage starts to accumulate.
This acid will etch and scrape into the clearcoat paint, then eventually it will work its way deeper and cause deep damage. This kind of damage can only be properly repaired by a paint shop.
Similar reactions happen with bird debris. The goop will sit and etch into the paint, causing weakening of the clearcoat.
Here's an example of what happens if you leave bird droppings too long:
In order to have long lasting paint that looks good, it is important to get any damaging material off of the paint quickly. With modern bug and tar remover products available at your local auto care center the job shouldn't take more than a few minutes.