Let 
$$ X, Y \subset \mathbb{P}^N$$
be two non singular algebraic varieties of dimensions $k$ and $l$ that 
intersect transversally. Is it true that the ``dimension'' of the variety
$\overline{X} \cap \overline{Y} - X\cap Y$ is strictly less than $k+l-N$, 
which is the dimension of $X\cap Y$ as a complex manifold. 
What I am worried about is that when you take the closure and then 
take intersections you may add singular things of very high dimension
to $X\cap Y$. 


I think it is true that the dimension of $\overline{X\cap Y}- X \cap Y$ 
is strictly less than $k+l-N$. 



If  you take the closure of two smooth varieties and then take their intersections, is the singular locus still small?
There are already two answers pointing out why your statement cannot hold as stated, so let's see if we can fix it. 


Let $X, Y\subseteq \mathbb P^N$ be two irreducible (quasi-projective) algebraic varieties of dimension $k$ and $l$ respectively. Then $\overline X,\overline Y\subseteq \mathbb P^N$ are two closed irreducible algebraic varieties of dimension $k$ and $l$ respectively.
By the Projective Dimension Theorem you obtain that 



> 
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> > Every irreducible component of the intersection $\overline X\cap\overline Y$ has dimension at least $k+l-N$.
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This implies that if your initial $X$ and $Y$ are disjoint, then your desired statement **cannot** hold. 


On the other hand since you assumed that $X$ and $Y$ intersect transversally, basically you only need to worry about the complements, that is, the interesting intersections are 
$\overline X\cap (\overline Y\setminus Y)$ and $(\overline X\setminus X)\cap \overline Y$.


**If** you know that these intersections are transversal, then I think what you want follows.


A perhaps interesting consequence of this is that if those intersections are transversal, then $X\cap Y\neq \emptyset$.


