Is a financial bubble harmless if rich people choose to hoard their wealth? The real question is longer than the title suggests. 

I was reading about Principles For Navigating Big Debt Crisis by Ray Dalio, and he mentioned that an economic bubble happens when too much money was chasing after too little goods. Effectively, it also implies that incomes should not rise faster than productivity. 

However, what if rich people (or those who own overvalued assets) don't spend their "profits" and hoard them? Wouldn't that mean the real money that was circulated in the economy will still remain stagnant, people's income did not actually rise faster than productivity, and that the bubble will just result in an arbitrary growing wealth gap that wouldn't have any real consequences until the rich decides to spend/distribute them?
an economic bubble happens when too much money was chasing after too little goods


This is not correct statement. When too much money chases too little goods you get inflation including asset prices inflation but asset prices inflation or inflation in general does not necessarily leads to bubbles. Bubbles are often thought to be behavioral in origin, see for example Minsky's Stabilizing an Unstable Economy or Shiller’s  Irrational Exuberance. For example Minsky argues that bubbles ultimately start whenever investors get excited about some new paradigm which can be some new technology or it could also be financial in nature such as shift to permanently more loose monetary policy but it is not the increase in money itself that creates bubble but rather the exuberance investors feel. There is no guarantee that just having too much money will do this. The Shiller's work has similar message. 

There can also be rational bubbles where the expectation of an asset price deviates progressively from the path dictated by its economic fundamentals  (see Nelson Mark's textbook international Macroeconomics and Finance) which could be caused for any number of reasons unrelated to the amount of money in circulation. 

Hence already the whole premise of your question is not generally valid as bubbles are not necessarily linked to the amount of money in economy. 

Furthermore, whether there is an bubble or not, just hoarding money can be harmless or detrimental to the economy but it is definitely not a positive. If rich people would just hoard money by not spending them or not reinvesting them then the money would become 'sterile' - it would be an equivalent of removing the money from the economy which would create deflationary pressure for the economy. 

Arguably, during boom such deflationary pressure is more or less harmless, although even then it would be better for an economy if the money would be invested into some productive activities instead of just hoarded. If an economy is in recession creating additional deflationary pressure is just doing extra harm to the economy as due to wage stickiness it will lead to increase in real wages precisely at times when they should be cut.  In addition, if there already is bubble on the market then the deflationary pressure would just make it burst not keep going (although arguably it might be better to burst bubble sooner than latter). Deflationary pressure at right times could help prevent asset bubbles but that would be only in cases where it is actually exuberance about inflation that causes the bubble and besides your question implies that the bubble already exists. It would also be unlikely that rich people would just 'hoard' money to create the optimum level of monetary contraction to prevent it so any such task should be preferably done by central bank. 

To sum up, hoarding money does not make bubbles harmless and it is not positive for economy. Hoarding money itself might be in some situation harmless to the economy but in most cases it would actually be detrimental in itself. Furthermore, by creating deflationary pressure it would actually make peoples real incomes raise faster than the productivity in presence of 'sticky wages' (at least in the short run until the deflationary pressure would become part of peoples expectations).