Donald Trump is by far one of the most sensitive presidents we have ever had and since he sings Putin's praises every chance he gets he might take offense to Russia allowing Snowden to stay in Russia and apply for citizenship. Russia may like Donald Trump but it's probably because they think he's a pushover. He's basically playing a game of show and tell where he shows you everything but he tells you nothing.
Trump is surprisingly open minded about why Snowden did the things that he did. Ultimately, Trump caught the intelligence community in leaking things to the press. Trump doesn't trust his intelligence community a whole lot more than Snowden ever did. Trump has bigger fish to fry than Snowden, and he'll leave him alone.
President Donald Trump will not likely be offended by Russia's refusal to turn over Edward Snowden. Trump never even mentioned Snowden during his campaign--the issue is a non-issue for Trump. The president will likely seek to cooperate with Russia when it comes to fighting the Islamic State, as well as terrorism around the world.
Trump won't bat an eye when Russia announces their plans to continue to host and safeharbor Snowden. Trump has already shown that he has ties with Putin and the Russian government, and isn't going to speak out against them. Whether or not they have blackmail on him is yet to be known, but Trump certainly seems to have an allegiance to Putin.