Is there a constant factor approximation algorithm for 2D rectangle coloring problem?
As suggested by the other answer, $\Omega(\log n)$ lower bound is not too hard to see. Let us do the sweeping bottom up with a horizontal line. The idea is to build components that require larger and larger number of colors. In particular, let $C(i)$ be a gadget that has a top rectangle with color $i$ (i.e., the first fit would assign it color $i$). Clearly, $C(1)$ is just a single rectangle. 
The component $C(2)$ is



In general, the component $C(k)$ is a rectangle with $C(1), \ldots, C(k-1)$ hanging below it:



Now, it is easy to verify that a fit-first algorithm with sweeping  horizontally from the bottom would use $k$ colors to color $C(k)$. However, the intersection graph of $C(k)$ is just a tree, and it can be colored by $2$ colors. Now, $C(k)$ is just a Fibonacci tree in structure, and as such the number of nodes in it is $2^{O(k)}$, which implies $\Omega( \log n )$ gap. 

Since there is a simple algorithm that gets $O( \log n)$ approximation to coloring of rectangles, this might be tight. I don't know.