It would seem to me, there is a difference between “opening for public use and possible prosecution” (my interpretation of the 4th amendment coverage) and opening sealed records for a president to look at them.
Sure Obama may be immune from prosecution, but I think Trump could look at whatever he decides to look at.
Not an expert however. Others may have more educated responses, but it seems to me, he can look at what he wants to look at.
A president does not have the authority to issue a search warrant.