Do dormant databases impact server performance?
Yes, you are right. Memory is used by dormant databases

I mentioned this in the following posts over the years


Nov 13, 2015 : #1041 - Out of memory issue in mysql
Apr 22, 2014 : Do Inactive MySQL Databases Consume Memory?
Apr 21, 2014 : Adding new tables -- memory usage increases


Lots of tables ? Lots of columns ? Of course, that is less memory for mysqld to work with.

You just asked


  Is there a way to ensure a database is using no memory?


If you never access a table, no data or index pages will ever touch the InnoDB Buffer Pool, the MyISAM Key Cache, or any session's read_buffer or read_rnd_buffer.

Notwithstanding, having too many tables with lots of columns will result in nothing more that a lot of dormant information_schema memory. Neither the number of rows in each table nor the size of any of the tables matter.