When creating a Test project on JEMS and indicating a duration of 33 months, JEMS automatically sets 6 project periods in the budget section. Does this mean that we must allocate some budget also to period 6 (months 31-33), although this is only dedicated to administrative closure?

Yes. The 6th project period in the budget section has to be fulfilled with an allocated budget and of course it may concern the administrative costs of project closure for example, or the last payments of implementation costs.