Suppose I have a (potentially non-linear) diffeomorphism $D:\mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$, consisting of functions $x_k' = f_k(x_1,...,x_n)$ with $k \in \{1,2n\}$, what concrete constraints must $D$ satisfy in terms of its $f_k$ if it is also a symplectomorphism? Could you provide an example for $\mathbb{R}^{2}$?
Please take into consideration that my knowledge of differential topology is rather limited.
Well, just writing the definition should be enough. If the symplectic form is
$$\Omega = \sum \Omega _{ij} \Bbb d x_i \wedge \Bbb d x_j = \sum \Omega' _{kl} \Bbb d x'_k \wedge \Bbb d x'_l ,$$
just write in coordinates that
$$\tag{*} f^* \left( \sum \Omega' _{kl} \Bbb d x'_k \wedge \Bbb d x'_l \right) = \sum \Omega _{ij} \Bbb d x_i \wedge \Bbb d x_j .$$
The left-hand side becomes
$$\sum \left( \Omega' _{kl} \circ f \right) \Bbb d f_k \wedge \Bbb d f_l = \sum _{k,l} \left( \Omega' _{kl} \circ f \right) \left( \sum _i \partial _{x_i} f_k \ \Bbb d x_i \right) \wedge \left( \sum _j \partial _{x_j} f_l \ \Bbb d x_j \right) = \\ \sum _{k,l,i,j} \left( \Omega' _{kl} \circ f \right) \partial _{x_i} f_k \ \partial _{x_j} f_l \ \Bbb d x_i \wedge \Bbb d x_j ,$$
whence by equating with the right-hand side of $(*)$ you get the necessary and sufficient condition
$$\sum _{k,l} \left( \Omega' _{kl} \circ f \right) \partial _{x_i} f_k \ \partial _{x_j} f_l = \Omega _{ij} .$$