County did take the lead in the 65th minute when a mix-up between Burch and Moses Swaibu left Delroy Facey to shoot into an empty net.
But Sam Mullarkey levelled for Lincoln just 30 seconds later, firing into the top corner of the net from 25 yards.
Lincoln City: Burch, Beevers, Kovacs, Swaibu, Brown, Kerr, O'Connor (Clarke 88), N'Guessan, King (Mullarkey 54), Wright, Patulea. Subs Not Used: Oakes, Horsfield, Miller.
Goals: Mullarkey 67.