The 29-year-old was carried off on a stretcher early in England's 47-17 victory over Italy on Feb. 14 after his head was struck by Andrea Masi's shoulder.
Brown took part in light training with the England squad on Tuesday but failed to pass Stage 3 of the return-to-play concussion protocol ahead of the trip to Ireland.
Lancaster said Saracen's Alex Goode will take his place for Sunday's match between the two unbeaten sides.
"We are, of course, disappointed to lose Mike but Alex is an experienced international and in great form for Saracens," Lancaster told reporters on Wednesday.
"Mike and our medical staff have been diligent during every step of the graduated return to play protocol and they will continue to work to get him back to full health. The most important thing is Mike's welfare."
(Reporting by Sam Holden; Editing by Ken Ferris)
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