$X, Y$ are continuous random variables taking values in $[0,1]$. Their joint density is $f(x, y) = x+y$ when $x, y \in [0,1]$, and $f(x,y) = 0$ otherwise.
Is it possible to conclude whether or not $X, Y$ are independent from just this definition without performing any calculations or mathematical manipulation? i.e. just by inspection of the joint pdf. If so, how?
On
The following lemma can be useful for such problems.
Lemma: A function $f:\mathbb{R}^2\to \mathbb{R}$ of two variables can be expressed as a product $f(x,y)=g(x)h(y)$ if and only if $f(x,y)f(x^*,y^*)=f(x,y^*)f(x^*,y)$ for any $x,y,x^*,y^*$ in $\mathbb{R}^2$.
Taking $x=y=0$ and $x^*=y^*=1$ we have $f(x,y)f(x^*,y^*)=0$ but $f(x,y^*)f(x^*,y)=1$, which shows that your $f$ cannot be written as a product function.
On
After getting some help suggesting the use of contour plots the correlation between $X$ and $Y$ became more clear. See below the original plot as in the OP (left) side-to-side with the color coded code (right):
which when seen from above does convey the idea of higher overall density in $Y$ as $X$ increases, although arguably not as obviously as on the case of a bivariate normal with correlation (left):
For the independence of these two random variables it is necessary that the joint density function can be decomposed into the product of the densities of these random variables. One can see that $x+y$ cannot be expressed in the form $x+y=f(x)g(y)$ for all $x, y \in [0,\,1]$.