The county where the warrany is active will transport him. In your friends case if he is in custody in Escambia County (Pensacola) then they would notify Tampa (Hillsborough County) and a transport deputy from Tampa will go pick him up and bring him back.
Although it is possible, Hillsborough County is unlikely to transport him on a misdemeanor DUI where the charges are stemming from Tampa, FL and he lives in Pensacola. It is more likely that the warrant remains active until he turns himself in on the warrant. Prior to him turning himself in to the Orient Road jail, I would recommend that you contact the Hillsborough County Jail to determine whether there is a bond assessed in his case to accelerate his release once he turns himself in.
This is not intended to be an attorney-client relationship and is my opinion only. Consult with a local attorney for additional advice.
They definitely will. It is required by law when within the state. It is better that he turn himself in at the Tampa jail or else they will impose additional transport costs too. Feel free to contact me as I handle Tampa cases.