Obama will speak out against Trump's policy changes but I don't think he'll be able to say anything that damages Trump's Presidency. I hope Obama takes on an active public role, even without holding public office. The Democratic party needs a sound strategy going forward and utilizing Obama's popularity should be an integral part of that strategy.
Obama has already been very critical and said unflattering things about Trump that he should have had the taste and decorum to keep to himself. There is nothing that hasn't already been said or that will damage Trump at this point. He will have to work hard to prove people wrong, but Obama has no more relevancy at this point.
Barack Obama is one of the classiest presidents we've ever had. One of his main objectives from the very start of his presidency was to bring this country together and I think that he will still continue to do just that and that includes supporting the president. With that being said I believe Obama would speak out if Trump did something wrong or unforgivable but he would do it in a respectful manner.