If $X$ and $Y$ are uncorrelated random variables, then under what condition is $E[X \mid Y] \approx E[X]?$ Suppose $X$ and $Y$ are real random variables that are uncorrelated. Now, uncorrelated does not imply independence, so $E[X \mid Y] \ne E[X]$.
However, can they be said to be approximately equal? If so, under what conditions does that approximation hold? (I realize that the equality is exact when $X,Y$ is a multivariate Gaussian, but I want to know more generally when the equality can be approximated.)
As a bonus, is it possible to do an expansion of $E[X \mid Y]$ that looks like "$E[X] + $higher order terms", so we can see explicitly when those higher order terms can be said to be negligible? Let's assume that $p(X \cap Y)$ is continuous and infinitely differentiable.

No, they cannot be said to be approximately equal in general unless they are exactly equal.   To see this, consider:
$$\mathbb{E}[X|Y] - \mathbb{E}[X] = \delta$$
for any $\delta$ that is near the boundary of what you consider to be "approximately equal".  Now, multiply $X$ by $10^9$:
$$\mathbb{E}[10^9X|Y] - \mathbb{E}[10^9X] = 10^9\mathbb{E}[X|Y] - 10^9\mathbb{E}[X] = 10^9\delta$$
and now $\delta$ is no longer in the range of "approximately equal" values.
You should be able to see this also prevents the expansion of $E[X|Y]$ from giving you any useful information in this regard; basically, one way or another, you'd probably need to actually compute the conditional and unconditional expectations and compare them to determine whether the difference is ignorable in your application, or perhaps compute bounds on the difference and use those as a decision tool instead.
