Can air inside a high temperature (1300C) Kiln cause an explosion?
I think in general the physics would make it unlikely that the air pressure would build up faster than it leaks out, attempting to seal a furnace is unlikely to happen by accident.

A bomb calorimeter or pressure vessel made to specifically seal the inside from outside does not have a loose fitting lid over a cast refractory edge, it is made to much finer tolerances.

Chemical reactions are not likely unless you have a significant mount of vaporisable fuel in the charge and this is contained until at some point it lifts the lid and lets vapour out and air in with glowing elements to ignite the mixture, a flame may result, explosion still unlikely as volume small and not contained any longer.  

In practice most kilns have a hole for temperature sensor that is not airtight and would let pressure equalise as they are heated. 

Unless a specific atmosphere (inert, reducing, oxidising, carburising) is required in a kiln there is no real need to make them air tight and they rarely are, even then it is often achieved within a protective wrapping of the load and supporting chemistry inside the wrapping like charcoal, cyanides etc).