Where the support is $x+y>0$. Taking partial $x$ partial $y$ reveals $f(x,y)=-e^{-x-y}$, but we know density must be non-negative. What is wrong here?
The CDF seem to satisfy all conditions of a CDF so it doesn't seem like there is a problem with the CDF itself. (1) $F(x,-\infty)=0, F(-\infty,y)=0,F(\infty,\infty)=1$ (2) F is increasing in both x and y (3) F is right continuous.
If $F$ is indeed a CDF and this of $(X,Y)$ then for $r>0$ we find: $$0\leq P(0<X\leq r,0<Y\leq r)=F(r,r)-F(r,0)-F(0,r)+F(0,0)=$$$$(1-e^{-2r})-2(1-e^{-r})+0=2e^{-r}-e^{-2r}-1$$
However if e.g. $e^{-r}=0.1$ then $2e^{-r}-e^{-2r}-1=-0.81<0$.
We conclude that $F$ is not a CDF.