The Premier League champions suffered a shock 1-0 defeat to the 10-man Lions courtesy of a late Shaun Cummings strike at the New Den, but the game was further marred by alleged abuse directed towards players and supporters.
A Foxes spokesperson told the Leicester Mercury: "We have personally congratulated Millwall on a merited victory and wish them every success in the sixth round.
"However, following receipt of numerous complaints relating to the constant abuse, provocation and intimidation of our players, staff and supports from our arrival at the stadium, throughout the match and in its immediate aftermath, we have registered our dissatisfaction with the Football Association.
"We are awaiting the FA's response.
"The club accepts the defeat, but we simply will not accept the safety of our supporters, players, and staff being compromised."
Millwall's reward for defeating Claudio Ranieri's side is a trip to Tottenham Hotspur in the sixth round.