Will World of Warcraft automatically upgrade to Burning Crusade and WoTLK

Yes, the software automatically updates itself when you log into the game. However, if you do not upgrade your account, you will only be able to access the areas you have paid for:
Vanilla - Old World, Level 60 cap
The BC- Outlands, Level 70 cap
WotLK - Northrend, Level 80 cap

The software will always be at the highest expansion, your account just has to be upgraded to use the expansion.