Is there a way to do a static route toward a DNS address instead of a
IP Address ? How ?
No, but perhaps what you want could be achieved with a local override in /etc/hosts.
One example of when this was useful from my experience:
I have a busy reverse proxy (on the "new" network), which fronts a number of sources. To get to one of those sources, the traffic would have to go through the old network which is desirable to avoid. However, if we added another interface (on the new network) to that backend host, we could make the traffic go through the new network. So in this particular case I just put an override on the reverse-proxies /etc/hosts that pointed to the new IP on the backend server.
(For Windows, just change C:\Windows\System32\drivers\etc\hosts)
Another example, this time with a browser proxy.
I was recently deploying a major upgrade to our website, and wanted to have people test things exactly as they would be (ie. same URLs as would be used post go-live).
So in this case I set up a browser/forward proxy (squid) on a new machine and altered its /etc/hosts to point to the new-world IPs, while leaving DNS to remain with the old-world IPs (until go-live). The effect was that if they configured their browser to use this proxy, it would go to the new-world website, otherwise they went to the old-world website. I also set up a PAC (Proxy Auto Configuration) file that had a set of rules to say which URLs should use the proxy, and anything else should go direct. Worked reasonably well.