What are natural examples of non-relativizable proofs?
As Steven notes, the canonical example is $\mathsf{IP} = \mathsf{PSPACE}$. This collapse does not relativize, in the sense that there is an oracle $A$, subject to which $\mathsf{IP}^A \ne \mathsf{PSPACE}^A$. The intuition why the known proof of this result avoids the relativization barrier is that it uses arithmetization (Yonatan alluded to this in a comment): an interactive protocol for the $\mathsf{PSPACE}$-complete problem TQBF is given by considering an extension of a quantified boolean formula to a low-degree polynomial over a suitably large field. If we are given a relativized boolean formula (with oracle gates), such an extension does not exist. 

There is a refinement of the relativization barrier -- algebrization -- due to Aaronson and Wigderson. Generically, the arithmetization technique is not enough to circumvent the algebrization barrier. A complexity class inclusion $\mathsf{C} \subseteq \mathsf{D}$ algebrizes if for any oracle $A$ and any extension $\tilde{A}$ of $A$ to low-degree polynomials over a finite field, $\mathsf{C}^A \subseteq \mathsf{D}^{\tilde{A}}$. A separation $\mathsf{C} \not \subset \mathsf{D}$ algebrizes if for all $A$, and all extensions $\tilde{A}$, $\mathsf{C}^{\tilde{A}} \not \subset \mathsf{D}^{A}$. Aaronson and Wigderson show that $\mathsf{IP} = \mathsf{PSPACE}$ algebrizes, but many other results, including $\mathsf{NP} \not \subset \mathsf{P}$, do not.

A recent example of a technique that does not algebrize or relativize is Ryan Williams' proof that $\mathsf{NEXP} \not \subset \mathsf{ACC}$. The separation does not algebrize: there is an oracle $A$ and a low-degree extension $\tilde{A}$ such that $\mathsf{NEXP}^{\tilde{A}} \subset \mathsf{ACC}^A$.  Intuitively the reason why the proof avoids the barrier is that it relies on the existence of a faster-than-trivial satisfiability algorithm for $\mathsf{ACC}$ circuits, and the algorithm uses non-relativizing and non-algebrizing properties of such circuits. Ryan notes in the paper that all known faster-than-trivial satisfiability algorithms break down when oracles or algebraic extensions of oracles are added.

There is also an interesting approach to understanding relativization through logic. In an old manuscript, Arora, Impagliazzo, and Vazirani define a system of axioms such that the relativizing results are exactly those that follow from the axioms, while non-relativizing results are independent from the system. A paper by Impagliazzo, Kabanets, and Kolokolova does something similar for algebrization by introducing an additional axiom to the ones defined by Arora, Impagliazzo and Vazirani. They show that most known non-relativizing results follow from their axioms, while P vs NP, among others, is independent of them.

Apologies if I got something wrong, I am not quite an expert.