Why do we need Cities of Refuge when Moshiach comes if at that time there will be no more death?
The Lubavitcher Rebbe answers that we will need the Cities of Refuge for all the people who have killed inadvertently over the course of exile. They will need to go to the city of Refuge to protect themselves from the "blood redeemer" (Go'el Ha-dam). See there for the explanation of why the "blood redeemer" taking revenge is not incompatible with the Days of Moshiach.
AskMoses.com adds that at the beginning stages of the Messianic Era (see Shaloh mentioned here), before the Resurrection of the Dead, the Cohen Gadol can still die, thereby freeing the accidental murderers from the Cities of Refuge.
The Aruch Laner answers that natural death will no longer exists but if someone kills someone death will still happen.
The Chasam Sofer in Toras Moshe on Parshas Shoftim says that maybe you can answer that the Geirim (proselytes) that come to Klal Yisroel will not reach the level of death not affecting them, but will be only counted as Jews and therefore the Cities of refuge will be for them.
The Shnei Luchos Habris known to us by his acronym as the Shlah answers there are many different time periods after Moshiach comes, the city of refuge will be needed for the initial stages and only later will death disappear from the world.
The Meshech Chachmah answers that there are two ways Moshiach can come we bring him with our good deeds or time runs out and history runs its course and Hashem is forced to bring Moshiach even though we are undeserving. In the case we are undeserving the outcome will be as the Rambam says the World will continue as it is accustomed and Death will still happen and that is why the city of refuge will be needed if we get Moshiach by History runing its course.