There is no hard and fast rule. It all depends on your queries. One bad query can take down the whole server and hang the mysql server. So being a shared server we have setup the limits at optimum level so that the server does not crash when one of the user site is hit with lots of traffic.
If you are getting "too many connections" mysql error, then either you are hitting the limits or your code is not closing open mysql connections before opening new ones.
Only person(s) who can tell you how many queries will make your site work fine is the developer of the code you are using. They should do a stress test and see how many queries will require how much system resources (cpu, ram). Depending on that result, we can help you with identifying which dedicated server will fit your needs. Otherwise it is just a hit and trial excercise.