Let $F$ be a non-archimedean local field with finite field $\mathbb F_q$ of prime characteristic $p$, and let $L$ be the completion of the maximal unramified extension of $F$. We write $\mathcal O$ for the valuation ring of $L$. Further, we denote by $\varpi$ a uniformizer of $L$. Set $G=\mathrm{GL}_n$. Let $I$ be the inverse image of the subgroup of lower triangular matrices under the map $G(\mathcal O)\rightarrow G(\overline{\mathbb F}_q), \varpi\mapsto 0$. Then we have the Iwahori decomposition $G(L)=\bigcup_{w\in \tilde{W}}I\tilde{w}I$.
My question is the following: if $w=w_1w_2$ with length($w_1$)+length($w_2$)=length($w$), then we have $Iw_1Iw_2I=IwI$?
So, I was finally able to figure this out and it is wrong in general. I will give a counterexample for $GL_2$, which can be extended to $GL_n$.
Let $$ w_1=w_2= \begin{bmatrix} 0 & \varpi \\ 1 & 0 \end{bmatrix},$$ then $$w=w_1\cdot w_2=\varpi\cdot Id$$ and furthermore $$l(w_1)=l(w)=0.$$ Thus your condition is met. Now we get an inclusion $$ IwI\subset Iw_1 I w_2 I$$ but it will be strict: Consider the matrix $$A=\begin{bmatrix} 1 & 1\\ \varpi & 1 \end{bmatrix}$$ which is in the Iwahori subgroup $I$. Then $$ B=w_1 A w_2=\begin{bmatrix} \varpi & \varpi^3\\ 1 & \varpi \end{bmatrix}. $$
Thus $B\in Iw_1 I w_2 I$, but it can not be in in $$IwI=w I=\varpi\cdot I,$$ as then every entry in the matrix has to be a multiple of $\varpi$.
Let me put this in a bigger picture: In general for $GL_n$ (you can actually also consider more generally split reductive groups, but then you have to be more careful) we have a short exact sequence $$ 0\to W_{af}\to \tilde{W}\to \mathbb{Z}\to 0$$ where the first map is the inclusion of the extended Weyl group of $SL_n$ and the second map is the ($\varpi$-)valuation of the determinant of an element in $\tilde{W}$. This sequence is actually split and decomposes $\tilde{W}$ as a semi-direct product. In particular we get a map $s:\mathbb{Z}\to\tilde{W}$ and essentially by definition for any $m\in \mathbb{Z}$, the length of $s(m)$ is $0$.
In the $GL_2$-case, the splitting is given by $s(m)=(w_1)^m$.