Duke in Irishman's chair (5)
I believe the answer is:
sedan
'chair' is the definition.
(sedan chair is a type of chair)
'duke in irishman's' is the wordplay.
'duke' becomes 'd' (abbreviation).
'in' is an insertion indicator.
'irishman' becomes 'sean' (common Irish name).
'd' going into 'sean' is 'SEDAN'.
(Other definitions for sedan that I've seen before include "Old carrying chair" , "American car" , "big chair" , "Palanquin; car" , "Saloon car in America" .)