Is it safe to leave cavity under crown?

If the tooth is indeed nonsavable and you are not prepared to proceed, taking the crown off becomes a problem.  If there is any chance of saving the residual root and placing a new crown, then getting the crown off and cleaning it up right away makes sense.  The danger of leaving a crown on until it falls off naturally is that you may swallow or aspirate it when it does drop off.  Talk to your dentist as it may be better to take it off if you are going to loose the tooth anyway.

If the tooth has large enough decay that your dentist cannot replace/restore it and the tooth must be removed then you do have some risks:1.  Infection2.  Bone lose3.  Tooth fracture4.  Pain/discomfortTypically you would not have to do a bridge or implant immediately after the tooth is removed.  To give you more possible time to save up, you can elect to have a bone graft placed in the area which will help keep the bone level higher and wider, this will give a better area for an implant.  The downside to waiting too long after you have the tooth removed is that there can be tooth shifting or angling as well as the opposing tooth of the other jaw may supra-erupt and begin to elongate into the open are where the tooth use to be.  This process does take some time and is different for everyone.

Since the dentist is saying that you would need a bridge or an implant, that means that that tooth needs to extracted or removed. So leaving the crown on for a while won't cause any more damage than is already done. The worse that could happen is that the crown breaks off or that food gets in around the gums and causes them to get irritated or infected. Keep it as clean as possible until you can have the tooth removed. Once the tooth is removed, you could look at having the dentist place some type of space maintainer or flipper to hold the space until you can afford to do a bridge or an implant. This would split up the cost for you and would still kepp your options open. Good luck