Electrons may or may not tunnel between the islands, there is no way of knowing without analysing the experiment in some detail (although this is such a common scenario that I am sure you can find papers where this has already been done).
However, heating and other thermal effects will under normal circumstances not really affect the tunnelling rate, the latter is usually temperature independent (unless the shape of the potential itself depends on temperature for some reason, not something I've ever come across), meaning heating should not really affect anything related to QM tunnelling.
That said, the heating would possibly affect the usual "over the barrier" rate, but this is just a classical effect.