WASHINGTON—President Donald Trump’s expected policy shift on Iran would put the GOP Congress in a difficult but familiar position: tackling issues that they opposed during the Obama administration but now look at in the new light of being the governing majority on Capitol Hill with a Republican in the White House.

Mr. Trump is expected to refuse to certify this week that Iran is complying with the 2015 multinational nuclear agreement negotiated by his predecessor. That would give Congress 60 days to decide whether to re-impose...