I was curious about what problems may arise if clutch is not used when shifting gears. Are the problems similar for both cars and bikes?
If you match revs correctly you should be able to change gear with no ill effects at all. The challenge is of course to match revs. This is pretty easy when accelerating, but when decelerating you'll need to use the accelerator to bump the engine revs up to match. Google for heel-and-toe (this also helps make for faster downshifts when using the clutch)
Getting started into first is tricky though. If your clutch does fail, parking on a down hill can help with this:-)
Excessive wear on the gears and a complete breakdown of the gearbox will be very likely to occur soon. Also the crankshaft and it's bearings will suffer a beating everytime you shift gears which will cause (fatal) damage to the engine.
The results will be similar (but definitely not the same) on a bike. If you keep pedaling while shifting gears you'll keep tension on the chain. This will momentarily put excessive sideways forces on the sprockets which will cause bent teeth and broken chain links.
If you must drive a bit when your clutch breaks down, for example on a railroad crossing, you can use emergency propulsion. You leave the car in gear and use the starter to drive the car for a few meters. Park the car at a safe location and call a tow truck.
There are synchro rings that match the speed of the gears when you push the stick. Synchro rings are not designed to accelerate/decelare the engine / whole car and will wear out quickly if you don't use the clucth.
...unless you are a real ninja with the gas pedal and match the gear speeds manually.
In a car with a conventional manual transmission, this will rapidly wear out the synchros unless you are able to match engine RPM to target gear RPM every time you change (possible, but not easy). Stopping and starting would be impossible to do smoothly without the clutch.
Many motorcycles use a sequential gearbox which allows shifting up and down without the use of the clutch (but you can only shift to the next gear in each direction). You still need to use the clutch for stopping and starting.