Question: I am not a mathematician. I have searched the internet about KL Divergence. What I learned is the the KL divergence measures the information lost when we approximate distribution of a model with respect to the input distribution. I have seen these between any two continuous or discrete distributions. Can we do it between continuous and discrete or vice versa?

Answer: 
No: KL divergence is only defined on distributions over a common space. It asks about the probability density of a point $x$ under two different distributions, $p(x)$ and $q(x)$. If $p$ is a distribution on $\mathbb{R}^3$ and $q$ a distribution on $\mathbb{Z}$, then $q(x)$ doesn't make sense for points $p \in \mathbb{R}^3$ and $p(z)$ doesn't make sense for points $z \in \mathbb{Z}$. In fact, we can't even do it for two continuous distributions over different-dimensional spaces (or discrete, or any case where the underlying probability spaces don't match).

If you have a particular case in mind, it may be possible to come up with some similar-spirited measure of dissimilarity between distributions. For example, it might make sense to encode a continuous distribution under a code for a discrete one (obviously with lost information), e.g. by rounding to the nearest point in the discrete case.
Not in general. The KL divergence is

$$
D\_{KL}(P \ || \ Q) = \int\_{\mathcal{X}} \log \left(\frac{dP}{dQ}\right)dP
$$

provided that $P$ is absolutely continuous with respect to $Q$ and both $P$ and $Q$ are $\sigma$-finite (i.e. under conditions where $\frac{dP}{dQ}$ is well-defined).

For a 'continuous-to-discrete' KL divergence between measures on some usual space, you have the case where Lebesgue measure is absolutely continuous with respect to counting measure, but counting measure is not $\sigma$-finite.