To get rid of DNS error code 80710102, the PlayStation Forum recommends bypassing your Internet service provider DNS server and use an alternative one. You need to access the Open DNS server to complete the process.
Open Internet Connection Settings
Go to the Settings tab on the PS3 and select Network Settings. In the Network Settings, select Internet Connection Settings.
Customize Internet settings
In the Internet Connection Settings option, select Yes and then Custom.
Select your connection method
A connection method can be either wired or wireless. If you have a wired connection, select Auto-Detect for the Ethernet operation mode. If you have a wireless connection, select the security type that your wireless router is using and enter your password. Your wireless security type can be WEP, WPA or WPA2.
Set IP address settings
In your IP Address Settings, select Automatic and then select Do Not Set for the DHCP host name. Set your DNS Settings to Manual and then enter the primary and secondary DNS numbers. The primary DNS number is 188.8.131.52 and the secondary DNS number is 184.108.40.206.
Enable Universal Plug and Play
Set the MTU to automatic, the proxy server to Do Not Use and then enable Universal Plug and Play. When you are done, click X to save the settings and then X to test the Internet Connection settings. This should get rid of the DNS error.