Do we need to update original/remaining time estimates when we estimate based on the story points?
In case if you don't use time estimates, why do you need to update these fields? Just don't use them.
If we estimate based on story points, then we need to look at the burndown chart based on story points and not remaining time estimates?
The same. If you don't use time estimates your suggestion is correct.
First of all, to not fit User Stories in a Sprint is bad practice. One of the main benefit of Scrum is fast feedback from customers. That's the reason, why business want to have "done" functionality after each Sprint.
About your problem: we use 2 level of estimation (you can read more here: Why use both story points and hours?) and use both metrics within Sprint Burndown Chart:
Yes, it was not well finished Sprint for us =)
- Red line is for tasks and represents man-days versus time.
- Blue line is for stories and represents story points versus time.
But in your case you can use only time-based metric (i.e. man-days) for tracking Sprint progress (yes, in that case you should start to estimate tasks). As for Story Points, use them for tracking something more high level (Release progress, for example).
P.S. Another solution (and it is much better): try to make your User Stories small enough to fit them to Sprints. This question "What we should do with tasks which estimated time bigger than sprint?" may help you to do this. After that you can use story-point-based Sprint Burndown Chart without any problem.