Am I going to get injured mountain biking if I don't have rear suspension? On the several occasions I've gone mountain biking, I was using a nice borrowed bike with full suspension and it felt nice.

I'm told that rear suspension, though nice, is not necessarily a must-have. I have notions of the bike seat bludgeoning the ol' undercarriage without it. What is the ride like without rear suspension, what do I lose? And are there any suitable substitutes or ways to compensate for the lack of rear suspension?
First off, when riding over rough terrain you should be hovering above your saddle, not sitting on the saddle.  This holds whether riding a hard tail (no rear suspension) or riding a full suspension bike. If you are sitting you have less ability to move your weight around, and therefore less control of the bike. When sitting it is easy for a large bump to throw you off balance making you crash or swerve into a crash.

When the trail is really steep and rough not only you should be off the saddle, but you should also be putting your weight behind the saddle.  Example below pictured below.  These riders are on downhill/freeride bikes (tons of suspension travel, 6-8 inches) and they are not sitting on the saddle.



So what advantages does rear suspension provide?


When the terrain isn't too rough (smoother than the above picture) it can allow you to sit down and pedal in more circumstances than on a hard tail, which can especially help when climbing (assuming an efficient pedalling suspension design)
Although your legs are the ultimate shock absorber (when hovering above the saddle) rear suspension supplements this natural suspension.  This can be critical when:


absorbing hits from big obstacles
landing off drops and jumps
absorbing a g-out
absorbing a series of fast rapid bumps as your legs may not react fast enough.
rescuing you from stupid mistakes or a bad line.

Depending on the suspension design, the rear suspension can also improve braking as the rear tire is in better contact with the ground.  (Although in some suspension designs braking can negatively affect rear suspension performance and therefore traction.)


What about hard tail?

To compensate for lack of rear suspension travel you need to use your legs more.  You should be riding above the saddle more often than with a full suspension bike.  For example, if you are pedalling over roots or rocks you need to learn to pedal while hovering over the saddle.  This lets the bike move over the terrain, maintaining traction and forward momentum.

So what are the advantages of a hard tail?

Hard tails are typically lighter bikes (less parts), and can be more efficient to pedal especially on long climbs. As such they still have a following among Cross Country (XC) mountain bike racers on smother courses.  That said, I have personally found the efficiency gains can be easily lost in rougher terrain (e.g., Squamish BC).  Hard tails are also cheaper to manufacture (the frame doesn't have any moving parts) so you can get a better equipped bike for your dollar.  This makes hard tails good first bike.

Hard Tail or Full Suspension

Whether or you choose hard tail or full suspension really depends on your experience, budget, riding style and roughness of your terrain.  

What constitutes rough differs from area to area. 

I personally live and ride on the North Shore of Vancouver Canada which has some incredibly rough and steep trails (pictured above).  My XC bike has 5 inches of travel front and rear and I use all of it on XC rides.  It does not have enough travel for some of the more advanced downhill trails in our area.

When I was visiting Bend Oregon, the riding was so smooth that my 5 inch XC bike was a rather silly choice and I would rather have been on a rigid (no suspension) bike.

In the end the choice is a personal one and the cost/benefit will be different for every rider.

Injuries

Injuries really depends on a combination of your bike handling skill, your riding terrain, and your risk tolerance.  There is also the probability of crashing combined with extent of injuries.  Full suspension often lets people ride harder terrain than they might otherwise, which has potentially bigger consequences. Anecdotally, I have observed those with the most suspension also tend also to have the biggest injuries.