The referees for the Premier League, Football League and FA competitions are appointed by the Professional Game Match Officials Limited. When referees make frequent and/or glaring errors in matches, the PGMOL reprimands them in the form of demotion to a lower tier of the Football League.
In the 2009-10 season, referee Mike Jones was demoted to the Championship after incorrectly awarding a goal to Sunderland against Liverpool, after Darren Bent's shot at goal deflected off a beach ball into the goal. Later in the same season, referee Mike Dean and assistant referee Simon Beck were demoted to the Championship following poor performances in a match between Manchester United and Chelsea. Dean failed to award a penalty each to both sides for tackles in the box and also allowed a goal to stand when he should have called a handball, while Beck failed to give an offside decision for Didier Drogba that led to a Chelsea goal.
In the game between West Ham United and Manchester United referenced in the question, only the second goal scored by Zlatan Ibrahimovic was offside. The other decision in the match that drew criticism was the harsh red card shown to West Ham's Sofiane Feghouli. It is up to the PGMOL to consider if these two decisions are significantly erroneous to warrant a reprimand against Mike Dean or assistant referee Simon Long.