There is always a performance hit when you increase data sources/number of viz's. But it's always dependent on how complicated the viz is and where the data is being pulled from. In your case with live connections you are relying on how well your data source environment is (memory, OS, disk speed), database efficiency, and finally network bandwidth.
If you find that it slows down considerably or even noticeably then why not use Tableau extracts? They are typically faster than a live connection especially if the database is not efficient.