Are the DRC errors in eagle software really important?
Yes, these are almost always serious, at least if you don't know why they occur.

My guess is that you have not changed your design rules according to your manufacturer but left them as default. For example, the default may have a 12 mil clearance limit while you are using tiny SMD circuits which requires an 8 mil clearance.

If you get actual overlap errors, then you may have drawn traces on top of other traces. Hard to tell without a screenshot and a list of errors.