26 U.S. Code § 6103(g)(1) says that the President can request tax return information from the IRS:
Upon written request by the President, signed by him personally, the Secretary shall furnish to the President, or to such employee or employees of the White House Office as the President may designate by name in such request, a return or return information with respect to any taxpayer named in such request.
This request, if made now, would have to be disclosed to Congress by Jan. 30, 2017 [by which point the election will (likely) be over and the next President inaugurated]:
Within 30 days after the close of each calendar quarter, the President... shall file a report with the Joint Committee on Taxation setting forth the taxpayers with respect to whom such requests were made during such quarter under this subsection, the returns or return information involved, and the reasons for such requests.
(A reason must be stated, but there are no clear additional requirements on what that reason may be).
§6103(a)(1) of the same title establishes the general rule that:
Returns and return information shall be confidential, and except as authorized by this title, no officer or employee of the United States shall disclose any return or return information obtained by him in any manner in connection with his service as such an officer or an employee or otherwise or under the provisions of this section. For purposes of this subsection, the term “officer or employee” includes a former officer or employee.
So it seems that Obama would not be able to directly share what he learns with the electorate, but merely disclosing that he has accessed the returns and after looking at them, really doesn't think Trump is in a competent position to take the office, might give more weight to what he's saying in at least some voters' eyes. Obama even publicizing that the option is there, with the implication he might use it, might add pressure for Trump to release returns voluntarily (limiting folks' imaginations), especially once the surprise factor of $0 paid (if that's accurate) has been absorbed.
Trump seems to be very willing to use executive authority to override treaties, international laws, etc. and thinks it's "smart" to gain maximal personal benefit from the laws in place in the tax code, so he'd seem to have a hard time challenging the ethics of Obama also using the existing tax laws to get the access the laws expressly permit him to have.
Trump famously pressed Obama for the release of his long-form birth certificate and still seems to think he deserves credit for getting to be able to see that, so Obama having a look at Trump's tax returns might be seen as at least partially a response which he can accomplish relatively easily from his current elected position.
Is there any polling, history, or other evidence indicating how the electorate would likely react to such a move?
Obama [...] disclosing that he has accessed the returns [...] really doesn't think Trump is in a competent positionwould be a breach of confidentality similar to directly publishing the data. To put an analogy, a doctor is not only forbidden of disclosing someone's medical condition but cannot either tell (publicly) something like "By the way, Mr. X, there is a new treatment which is really effective at healing ghonorrea, after seeing the results of your last analysis I relally think you should give it a try".