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.

We have to change one generator $\begin{pmatrix} 0 & \pi \newline \pi^{-1} & 0 \end{pmatrix}$ to $\begin{pmatrix} 0 & \pi \newline 1 & 0 \end{pmatrix}$, precisely for the matter you mention in the comments. Note that $\begin{pmatrix} 0 & \pi \newline 1 & 0 \end{pmatrix}^2 \in Z(F)$ of $GL(2)$.

I have not played with $n >2$ so far. But probably for $GL(n)$, something similar is possible.

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 pasting the center approach. 

Of course you can also look at $PGL(n)$, but this feels slightly different than caring the center along. The building should give you some control about the representation theory of the group in question, and there are certainly smooth admissible representation of $GL(n,F)$, where no twist has trivial central character.