Mechanism of lithium-halogen exchange of primary alkyl iodide
With primary alkyl iodides the (simplified) mechanism is likely similar to a $\mathrm{S_N2}$ process, but with attack of $\ce{t-Bu}$ at iodine instead of carbon.



Note that this is not necessarily the case for other substrates, e.g. aryl iodides, or aryl/alkyl bromides. In general lithium-halogen exchange can occur either via a polar mechanism (such as the one shown above) or via a radical mechanism (by two successive electron transfers from RLi to RX). There is also a solvent and temperature dependence. Why does the solvent matter? We know that ethereal solvents such as Et2O or THF are capable of acting as Lewis bases - or ligands for lithium - and possibly assist in deaggregation of the alkyllithium. This sort of thing can't be depicted easily with curly arrows, which is why I add the disclaimer that the mechanism above is (heavily) simplified.

In any case, this particular substrate (Tetrahedron Lett. 1986, 27 (17), 1861–1864) is fairly interesting because if the Li/I exchange were to occur via a radical pathway, the radical formed would cyclise onto the alkene via a 5-exo-trig pathway (even at −78 °C):



However, if you quench the reaction at −78 °C with methanol, no methylcyclopentane is observed, which rules out the intermediacy of a radical. [If you warm up the reaction before quenching then the organolithium can cyclise by nucleophilic attack onto the alkene.]

This particular case (and general mechanisms for Li/Hal exchange) are described quite thoroughly in Clayden's Organolithiums: Selectivity for Synthesis, Chapter 3.

There is also an excellent set of slides on Li/Hal exchange reactions, courtesy of the MacMillan group at Princeton; and also another set, courtesy of the Denmark group at the University of Illinois.