If my drone crashes and I want to dispose of it, how can I do so safely? Is it necessary to disassemble the drone into different parts: motors, batteries, propellers, shell etc. ?
In the UK a drone would almost certainly be considered WEEE waste (Waste Electrical and Electronic Equipment.)
You can take WEEE waste to your local waste handling/recycling facility; alternatively, there are "Take back Schemes" where some distributors will accept WEEE waste. This is often geared more towards white goods (fridges, washing machines, etc.); unfortunately you will need to check locally.
The best option is to break it down and separate the parts into the following:
Once the drone is broken down into these items, contact your local waste management and recycling centers to determine how to hand the parts over for proper disposal.
Note that some of the parts may still be in good working condition, so if you no longer want the parts, someone else might be happy to buy them from you. Selling used, but working, parts is a great way to reduce waste and still recover some of your investment while at the same time, allowing someone else to buy working parts at a reduced price.
A bit of a frame challenge here... but you could try to sell it for parts or give it away at no cost to someone who wants to try repairing it. This keeps it out of landfills / recycling depots, and may help others as well.
There are all sorts of websites around on which you can list things like this (Craigslist, Kijiji, etc).
In most cases, you're going to do everyone a huge favor by removing the liPo batteries and taking them to a place where they can be recycled or disposed of safely. In the US, there are many locations that will accept lipo batteries. Call2Recycle is probably the largest (they work with many major retailers that sell liPo batteries).
As far as the rest... that varies wildly across cities and states. Many states do not have electronic waste or recycling available (in some areas you might even have to pay for disposal). DJI has a recycling program which covers phones and tablets as well, which can garner you credit towards a new DJI purchase (sadly, it seems you have to buy a new drone from them to get the recycling of your old one). There are other places that say they will buy old drones, but that's probably a refurbishment program to resell on the secondary market. Consult your local waste disposal company or city waste department to find out what options are available for simply trashing one.