I am flying from from Phoenix to Dublin and I have a connection in Newark. My connection is in the same terminal. Terminal C. Will I have to go through security again?
On the outbound flight you will not need to re-clear security in Newark. You will also not need to clear immigration as the US doesn't have (physical) outbound immigration. You will simply need to go directly from your inbound gate to your outbound gate.
If you are flying the same route on the return flight then it will MAY be different, but probably will not be. Dublin airport is one of the few international locations where you go through US customs and immigration at your departure airport (ie, in Dublin) via a process called "Pre-clearance".
Presuming your flight goes through pre-clearance in Dublin then the process on your return will be the same - you will deplane directly into the departure area of the terminal and can go to your gate. If your flight does NOT go through pre-clearance in Dublin (not all flights do) then you will need to go through immigration, customs and security in Newark before catching your connecting flight.
No, there's no need to go back through security when traveling domestic within the US.