The tie will take place over the weekend of January 5/6.
The Toffees were ball number 15 in the draw, which was shown live on BBC 1 from 7.30pm.
It will be the first time the two sides have met since October 1993, when the Blues triumphed 4-2 in a League Cup tie.
Lincoln are currently second in League Two, and beat Northampton Town and Carlisle United to make the third round.
The Blues have won the competition five times in their 140-year history, but their FA Cup triumph over Manchester United in 1995 remains their most recent piece of silverware.