  We develop a general theory for the existence of extremal K\"ahler metrics of
Poincar\'e type in the sense of Auvray, defined on the complement of a toric
divisor of a polarized toric variety. In the case when the divisor is smooth,
we obtain a list of necessary conditions which must be satisfied for such a
metric to exist. Using the explicit methods of Apostolov-Calderbank-Gauduchon
together with the computational approach of Sektnan, we show that on a
Hirzebruch complex surface the necessary conditions are also sufficient. In
particular, on such a complex surface the complement of the infinity section
admits an extremal K\"ahler metric of Poincar\'e type whereas the complement of
a fibre admits a complete ambitoric extremal K\"ahler metric which is not of
Poincar\'e type.
