Could I kick him out for hating Trump? I know I cannot discriminate on religion, but politics? no law in that!!
What does the Obama supporter look like? If he appears to be a 95%er you could have a problem. As long as you treat all potential tenants the same, it's hard to prove discrimination. If you want to eliminate the irresponsible, require a minimum credit score of, say, 750. To eliminate potential trouble require an authorization to do a law enforcement background check. Ask all applicants if they have ever filed charges in or had charges filed against them in landlord-tenant litigation.
In your for rent ad, put "No liberals need apply" (just kidding) describe the dwelling and the location generally, but give out no address; require that they submit an application first and only show to qualified people. You do not want to meet them until you have qualified them, so they can't claim racism because you knew what their race was.
To see how they keep their current rental, go pick them up for the showing if that is practical, or at least do a drive-by.