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.
I am not totally sure about $n >2$, but I would guess you can take the usual building for $SL(n)$ and exploit $$ Z(F) \cdot SL(n,F) = GL(n,F).$$
Large center is pretty annoying sometimes, but does not really introduce new phenomena.
Another cheap trick is to work out the theory for $$ GL^1(n,F) = \{ g : | \det g|=1 \},$$ which has compact center, but I prefer the modulo center approach.