Q. Does a son inherit the seat acquired by his father, or assigned to him as a tribute to his eminence, after the death of his father? (See question 2517 above).
A. Poskim explain that the Shulchan Aruch (Y.D. 240: 2), seems to infer that calling one's father by his first name, is prohibited even after his demise. However, seating on his seat is permitted after a father's death. (R' Ch. Plagee in Chaim Beyad Y.D. 125: 48, Miyaam Hachalacha 2: 20).
Horav Shlomo Miller's Shlit'a opinion is that as far as the seat acquired by his father is concerned. it will mainly depend on the conditions and stipulations that normally the shul maintains for seat acquisitions.
Regarding the seat assigned to his father as a tribute to his eminence as a Talmid Chacham or as a tribute to his distinction in any other areas, it will depend if indeed the son fulfills similarly those same conditions.
Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit'a