Yes, of course, Jira will let you save a task with the 'Start date' and 'End date' fields empty.
Now, what effect will it have on what you see in BigPicture? It much depends on what you had set here: Jira administration > Manage apps > BigPicture > Technical configuration > General > Task > Task configuration
With the commonly seen configuration, evident in Fig. 1, the BigPicture Gantt chart, for instance, will show a newly created task with Start and End dates empty as a very "narrow", only one-day-long bar.
On the other hand, if you had made the setting shown in Figure 2 (-- Not Synchronized --), then if you leave Start and End dates empty on a task then it will have no effect on the bar on the Gantt chart; BP will expect you set the bar length "manually", by dragging&dropping it.
Still on Fig. 2, note that you can make this Start/End date setting per each project, separately.
More scenarios are discussed here: https://wiki.softwareplant.com/display/DOCUMENTATION/Task+configuration#Taskconfiguration-TaskPeriodConfiguration
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