Burch did well to keep out an early shot from Colin Daniel and made a string of fine saves throughout as the Silkmen dominated the first half.
When Burch was beaten five minutes before half-time by a header from Nathaniel Brown, Joe Anderson cleared the ball off the line.
Lincoln only really sparked into life in the second half. Steven Lennon put Lenell John-Lewis clear, but the striker's shot was blocked. Davide Somma fired against the post after 58 minutes and from the rebound Lennon sent his follow-up shot against the opposite upright.
The home side were then reduced to 10 men after 66 minutes when Somma was shown a straight red card for an off-the-ball incident involving Sean Hessey.
Lincoln had their best chance six minutes from time when Brian Gilmour played in Matthew Saunders, but the midfielder's left-foot shot was saved by Jose Veiga.