...lawn, will they eventually die out? Foolishly, I let the chives go to seed. Now they are coming up all over. I have been digging the little bulbs, but there are thousands of the little things. I can't use pesticides. It would be easier to keep cutting them off above ground, at least in the grassy area, as it gets mowed every week, but don't know if that will stop those little bulbs from sending up new shoots. Do I just need to keep digging?

YOU have to dig chives out by the bulblet/root
Cutting them down only makes the root stock stronger I have found.
I had planted both pink and white flowering chives in Northern New England. . . So much for tender perrenials, I cannot get rid of them now with out a pitch fork!

Not likely.. they are a bulb they will keep going... but why get rid of em? just make a salad... Most people try and grow these things on purpose..