damaged microwave door

To answer this question a bit of physics comes in handy:

Microwaves emit, well, microwaves, this is a form of electromagnetic radiation, like light or heat, with wavelengths (basically the distance between two peaks of the waves) between 1 meter (39.37") and 1 millimeter (0.039") and frequencies (number of times a wave goes up and down per second) between 300 MHz (300 million/second) and 300 GHz (300 billion). Interestingly, these wavelengths are pretty similar to those used in the radio frequencies - like WiFi. These wavelengths are very strongly absorbed by water, which is why it you can easily boil water in a microwave and why they are touted as "cooking from the inside out".

To stop the microwaves coming out of your microwave oven, it contains something called a Faraday cage. Faraday cages are basically a way of blocking the microwaves from coming out of your oven and cooking you or anything else nearby. In your microwave the cage will look like a wire mesh covering the door - the rest of the microwave is covered in sheet metal, which works the same way, but you can't see through it, and people like to be able to see into their microwaves.

At a minimum for a Faraday cage to work it needs to have have holes smaller than the shortest waves it is trying to block - in the case of microwaves this will be 1 mm. If the hole that the pan generated is bigger than this, then it can emit some microwaves.

Now the part we can't answer is how much radiation is getting out and how will this affect you - these can only be determined by measurement of the microwave emissions from the microwave. I recommend that you get it checked by a competent electrical service.