There are two possible meanings:
1. "I don't want to burst your bubble, but ..."
(= I don't want to disappoint you, but... [some disappointing news])
2. "I don't want the bubble to burst." This could refer to an economic "bubble," for example the price of houses in the United States (the "housing bubble"). The price went up and up and up, but then suddenly the bubble burst -- after that prices went down quickly.
So "I don't want the bubble to burst" can mean "I don't want the prices to fall."