Both were eligible to enter qualifying thanks to a scratch handicap, but on this evidence will not be rivalling the tour professionals in their adopted sport.
Murphy, ranked 14th in the world on the green baize, carded a 12-over-par 84 at County Louth.
"It is all a bit tongue in cheek and a bit of a laugh," Murphy told the BBC earlier this month.
"The one thing I do have going for me is that I already have a full-time job. Golf isn't my livelihood. Relative to the other players I will be playing against, it doesn't really matter to me if I win or lose."
Ervine, a former all-rounder for Zimbabwe, fared a little better but was still well off making it into the final round of qualifying with a five-over-par effort at Frilford Heath.
The 148th Open will take place at Royal Portrush from July 18-21.