What permissions do I need to request a DBA to synchronize a SSAS 2008 cube from a server1(where I'm the administrator on windows ) to a new server which is maintained by a different team where I will not be provided any admin rights?
To setup synchronization between two SSAS servers you need to have a common domain account with privileges on both SSAS servers, specifically you have to be SSAS administrator.
For further info, have a look at this step-by-step article https://www.mssqltips.com/sqlservertip/2938/how-to-synchronize-two-ssas-servers/