TubePress requires PHP5 or newer version. So if you are still using old version of PHP you will need to upgrade to PHP5 to be able to import youtube videos to wordpress. You don’t have to worry about this task so much, most hosting companies have both options PHP4 and PHP5 so you can choose yourself PHP5 from the control panel. If you are on some shared hosting that do not offer PHP5, you can just ask them to upgrade it for you, and they should probably do it for a couple of bucks. The newer version PHP6 will be soon released anyway.
On the other hand, any type of mysql server later than MySql 4.0 should be ok.