Usually one talks about the generalized BN-pair, when you have non-compact center. I am not sure exactly what the definitions are, but here is a cheap trick for $GL(2)$.
Enlarge the Iwahori subgroup by the center $$I = B \cdot Z(F),$$ then you get a nice BN pair.
Note $\begin{pmatrix} 0 & \pi \newline 1 & 0 \end{pmatrix}^2 \in Z(F)$ of $GL(2)$, and so forth.
Large center is pretty annoying sometimes, but does not really introduce new phenomena.