No, they can coexist with overlapping, concurrent validity. You do not need to revoke the old certificate in order to obtain the new certificate, and the new certificate will not invalidate the old certificate.
Depending on your authorization method, you will need to point the DNS records at the new server in order to obtain new certificates there. If the DNS change is authoritative, you wouldn't need to wait 24 hours. (Let's Encrypt doesn't rely on DNS caches and so the "DNS propagation" delay that people often experience doesn't apply to Let's Encrypt's infrastructure, although there could be internal delays inside DNS providers in applying the changes that you request.)