To test transistors you can short the base to the emitter and the collectors voltage will be close to +VCC supply. Can I do this to any transistor that is an amplifier or will this damage the next following transistor or previous transistor stage?
Can I short circuit a tubes cathode to grid to test a tube stage or short circuit a FET from gate to source to test a FET stage?
To test transistors you can short the base to the emitter
For sure you can do this with all transistors. But this configurazion is only useful fir testing particular parameters.
You can not test the complete fuction of a bjt with this setup.
or will this damage the next following transistor or previous transistor stage?
You can use e-b shorts for BJTs but first you must know circuit schematic to be sure this short cannot kill other components. Do not try this with vacuum tubes, they will open and glow yellow! Similarly with FETs.
Learn electronics basics, your daring tests can be costly.
There are many tests for transistors. Shorting the base to the emitter simply checks that it is not shorted from collector to emitter and does not test its current gain because the transistor is completely turned off.
The previous stage might be destroyed if you short the base to emitter but the following stage might survive.