Players from both clubs were involved in a heated exchange at half-time during United's 2-1 win on Saturday, while Evra was spoken to by referee Phil Dowd after the final whistle for his overt celebrations in front of Suarez.
The FA say the refusal to shake hands is not a disciplinary issue, while Dowd has reported that he dealt with Evra at the time and that the tunnel incident was not serious enough to warrant any further action.
Suarez apologised on Sunday for his refusal to shake the hand of Evra, the target of his racist remarks in October which led to the Uruguay striker receiving an eight-match ban.
The pre-match handshake routine takes place before every Premier League and FA Cup match and is part of the FA's Respect campaign.
The Premier League had ruled out scrapping the routine for the match, the first time Evra and Suarez have come face to face since the original incident in October, saying to do so would negate the point of the handshakes.
Statements from Liverpool manager Kenny Dalglish and managing director Ian Ayre on Sunday confirmed Suarez had misled them by indicating that he would shake hands with Evra.