This error does not sound like a problem with the password at all, but rather a connectivity issue. It could be that some policy is re-establishing firewall rules, or it could be simply that you have flaky DNS, an unreliable connection to the domain controller, or some other network issue.
As @Thomas pointed out, the "pipe" in the error message implies that this is a named pipes error. In most cases I have tended to avoid named pipes altogether. On the server where the SQL Server instance resides, you should go to SQL Server Configuration Manager and experiment with disabling named pipes and making sure TCP/IP is enabled:
Then you'll want to make sure that your app's connection string uses an IP address if feasible or edit the hosts file so that the name resolves to an IP locally, instead of using a physical machine name or FQDN. Doing either of these eliminates your dependency on external name resolution from a domain controller (but obviously won't be a good idea if you are using DHCP for some reason).
Of course you may also want to review network logs (or set up some kind of network monitor) to determine if there is some issue outside of this, and make sure that neither machine has a firewall enabled (or at least ensure that it is open on the right ports and no group policy is coming in and "fixing" that). But these changes will be better for your environment even if they don't solve this immediate issue.
As an aside, you absolutely must fix that application's connection string anyway. A blank
sa password is like leaving your wallet at the bus station on purpose.