What is the future role of tester?
IMHO that depends on what the role of the tester was before such transformation. BTW, I believe my answer applies to the DevOps transformation in general, not only to the You build it, you run it paradigm.

If the role was a drone job - mindlessly executing tests - it's bound to go away, automation will eat such jobs. 

If the role included writing the test plans and specifications they could continue to do that or even morph into writing the corresponding automation scripts, side-by-side with the developers. Or standalone - in some environments it is mandatory to use a completely separate methodology/infrastructure/personnel for testing than the one(s) used for development. In many organisations testers are software engineers, just as developers.

If the role was executing intentionally unreliable/surprise tests, specifically requiring human traits (emotion, subjectivity, inspiration, reflexes, intuition, etc) to complement automated testing - that also won't change.