The way I understand the idiom, you "break into" a cache of something that you might have set aside for a circumstance other than the current one, like savings accounts, or some bottled drinking water set aside for a future camping trip. I'm not sure of the etymology, but I always thought it came from the type of piggy banks that you had to break to access the coins inside. I always associated it with "all or nothing" and I would use "dip into" if I were talking about only taking some of my cache.
There is a lot of variability in how different folks interpret the same idiom. I wouldn't for example say "break into a new jar of peanut butter", I would say "crack open a new jar". I wouldn't think it was really strange if someone else said "break into a new jar". I would understand that they use the idiom differently than I do.
You could say "We will have to break into some of our long term investments to buy a house." If you aren't going to liquidate all of your investments, you might say "We will have to dip into my retirement fund to buy a house." I agree with TRomano that when referring to specific investments, like gold or stock shares, a native speaker would use a different more specific phrase like "liquidate" or "sell".
You wouldn't say "I will have to break into my paycheck this week to pay my rent." because typically your entire paycheck isn't set aside for something other than paying for your expenses. You might say "It's 2 AM and I really want some candy, so I'm going to break into the chocolate I bought for the Christmas party and go buy more when the market opens."