The likelihood of ash running out of the fire pit in the rain is going to be determined by how deep the pit is, how the bottom is treated, and what your soil is like.
If you have very porous soil with good drainage, then all but the most torrential of downpours will probably let the water soak into the ground without flooding out the fire pit.
If, on the other hand, you've got heavy, dense soil that drains poorly in general, or you've made a non- or limited-porous lining on the bottom of the fire pit, then your chances of having ash laden water flood out of the pit are increased.
If the pit is very shallow, then a good rain is likely to flood out of it no matter what you do. If it's fairly deep and you have otherwise good drainage, you may end up with a bit of a swimming pool, but it should drain out the bottom on its own, instead of overflowing.
If you're getting very heavy rain and the rest of the neighborhood is flooding too, then there's nothing much you can do to prevent your pit and whatever may be loose in there, from overflowing. At that point, though, you've probably got other things to worry about. At that point, there's probably enough water leaking in even with the lid on, that the pit may be filling and might just float the lid off, if it's light enough.