What happens when sequences/inboxes are paused for some time?

Follow-up schedules will still continue counting even if the inbox or sequences are paused.
That's why pausing sequences or pausing the inboxes may result in accumulation of pending follow-ups.

The longer the inboxes/sequences are paused - the more follow-ups may accumulate (especially if you're running multiple sequences with multiple steps). 

It's recommended to check how many follow-up emails are on pending before resuming the sequences. 
You can check that by going to Sequences -> Sequence List and check the pending follow-ups column. 
The number of pending follow-ups will be in red. 
See example: 

If the number of pending follow-ups exceeds the number of your average daily email volume, then it may cause problems if you resume sending immediately due to the sudden increase in email volume (especially with GSuite and Outlook email accounts). 

There are multiple ways to moderate the email volume so that you won't run into issues with your email provider.
Here are some suggestions:
1. You can reschedule the follow-ups - How to change the follow-up date/time
(using filters on the Prospect List page will help)

2. You can unpause one sequence at a time and only let a reasonable amount of follow-up emails to go out each day (while keeping the other sequences paused).

3. You can increase the delay between emails in your inbox settings. 

You can do all three suggestions above to ensure that your email volume won't surge out of control.

I also suggest not to add new prospects to your sequences until all pending follow-ups have been sent out.