Let's pretend Jira isn't the issue for a moment.
First, tasks should be estimated in hours, not points.
Second, stories should only be counted toward the burn down when they're complete. (https://www.scruminc.com/sprint-burndown-by-hours-or-by-story/)
Third, consider what value you are getting from the tasks. In teams where we have done tasks, I've found the work of creating the tasks a great validator of whether or not the story has been defined well enough. Marking off the tasks has been of less use to my teams.
Fourth, when a story is not complete at the end of the sprint, it should go back into the backlog for the product manager to re-prioritize. It may be that it's no longer a priority (especially if it's taking much longer than expected).
https://pm.stackexchange.com/questions/3990/unfinished-stories-in-a-sprint
From teh scrum guide - "All incomplete
Product Backlog Items are re-estimated and put back on the Product Backlog."
Personally, I don't like to re-estimate because then some effort is "lost". If you don't re-estimate, then your velocity balances out over time, even if each individual sprint velocity bounces up an down.
In some cases, when stories tend to be large, when they are estimated to take more than 4 or 5 days you can easily lose track of progress within a sprint. This is especially true if the sprint is 2 weeks long.
Teams I've worked with have moved away from tasks in favor of smaller stories. (Currently, we use a rule that if it's > 13 story points, it's probably too big and should be split up).
All this considered, Jira should fall in line fairly well.