It depends, quite often they are fine if touching the ground. When they touch the ground there is a higher chance of infection of some sort. If they are barely touching I wouldn't worry too much. If they are pushing hard into the dirt, a slight pruning might be in order, just far enough back to clear the ground though.

No, they will not die if branches touch the ground. And the remote possibility of any disease that may arise will be more likely due to the density of the branching structure and lack of air circulation than it is from any ground contact. There is the possibility of some foliar damage, but primarily from mud or rain.

I've seen scores of mature weeping laceleafs with branches sweeping the ground - it's a very graceful look. My own 'Green Cascade' Acer japonicum has this exact habit with many branches actually extending along the ground.