Following on the first response, you need a dog. When the lights blink, he is trained to bark and you know someone is at the door! Simple, eh?
Seriously, there probably is some kind of a short in the line. It is, as noted in the first response, not enough to blow a fuse. Actually I believe there was an inline fuse in the doorbell circuit, near the bell itself, and it may have been replaced with one which is too large and will not blow.
Since it happened after you replaced the bell button itself, I would guess the problem may be in or near the new doorbell button.