Is a "T" (Tango) prefix required before the registration number for Part 135 flights?
Since you're asking if it's required then I believe the answer is no.

First, just to include the information from another answer, the AIM 4-2-4 only says "should", not "must" (my emphasis):


  4. Air Taxi or other commercial operators not having FAA authorized call signs should prefix their normal identification with the phonetic
  word "Tango."
  
  EXAMPLE- Tango Aztec Two Four Six Four Alpha.


Second, the FAA's Joint Order defining ATC regulations says in section 2-4-20 (my emphasis):


  Air taxi and commercial operators not having FAA authorized call
  signs. State the prefix “TANGO” on initial contact, if used by the
  pilot, followed by the registration number. The prefix may be dropped
  in subsequent communications


That's the only mention of this prefix in the entire ATC regulations, therefore it's safe to say that controllers don't need the prefix and aren't required to do anything when they hear it.

On the AOPA forums there are a couple of unsupported comments that the prefix was used to give commercial flights priority in the past, and that it's still used for gathering statistics on flights today (although it seems like a very unreliable mechanism for that).