Although creative, I doubt that this is the best idea ever. I don't think this is possible and - even if it were - you would run into so many problems that I'd recommend other solutions:
If your goal is to incentivise logging work, make sure you make this visible. For team leads to be able to follow up team members' timesheets, for team members themselves just to see the status of their timesheets (potentially compared to their team members'). Project the team timesheets on a big tv, take a minute or so to ask people if they have completed yesterday's timesheet, ...
There's a lot of ways you can do this, using JQL filters or using timesheet add-ons like Tempo Timesheets.
I completely agree with Walter here, this would cause you all sorts of problems and really annoy your users. Inform them, don't beat them up.
I wanted to add the technical point too - you would need to write code, to do this. I don't think you could even do it in an add-on, I suspect you've have to hack the core of the permissions system.
That also means it's completely impossible to do on Cloud, as you can't touch the code.
I suppose you are referring to how the system may do this automatically for you. We do this with Tempo Timesheets and use the included Scheduler to send out notifications on a monthly basis (this can be weekly or bi-weekly as well).
On a daily basis, we have our standup meetings and a one of our colleagues who shouts out whether yesterday's timesheets are complete with everyone - I kid you not. May seem a bit strange, but fairly effective and less intrusive than sending a daily e-mail or something, which very soon will end up being ignored as it is not personal anyway.
Try to explain to your team what the importance is, communicate a lot and at regular intervals close and approve timesheets. That works pretty well in our daily practice.
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