Lazarus pits do not make their users immortal. Lazarus pits repair any damage done to the body, and magically re-initialize the life force, as long as your body isn’t too damaged, decayed or suffering from terrible brain injuries.
You can seem almost immortal to people who see you regularly and realize you never get any weaker, you don’t appear to age, and your physical capacity remains at peak levels for decades.
Technically as long as you can find them, you are functionally immortal but aging just the same. However, there are a few, uh, provisos, a, a couple of quid pro quos.
Lazarus Pit - use at your own risk

The best answer in the use of the Lazarus Pits is this:
- You will be insane when you are resurrected. Prepare for that contingency.
- You will have superhuman strength for a time. Have stronger friends.
- The more recent your death, the fewer issues you'll have upon resurrection.
- Try to die with as few injuries to the brain as possible. More risks involved.
- Make sure you are all there when you are put in the pit. No clues as to whether regeneration of missing limbs occurs.
The mental state of the person revived may be partially related to how they died, what kind of trauma, what they died from, their state of mind upon death to name just a few. For example: Ra's al Ghul knows, if he dies, the pit will bring him back to life. He is not afraid of the pit or its consequences. His strong will, sense of purpose and personal drive ensure he will be resurrected with only a minimum of damage, relatively speaking; he is after all, in his own way, quite insane. Brilliant, but insane.
How it works:
The Lazarus Pit's effect is essentially a magical one; powered by a confluence of ley lines, i.e. magical energy lines of force crossing the planet, which, when they are in a proper conjunction create a difficult to duplicate effluvia that has the ability to restore a person to life, along with reconnecting the soul to the body. Aside from the insanity, it appears to be a perfect resurrection vehicle. It effects been artificially created more than once, but it seems to require both significant technological resources, and some magical ability, understanding and resources as well.
While little has been shown as to its efficacy for the long dead (i.e. years dead) there was an attempt to raise Bruce Wayne's family and no one considered it unreasonable to expect it to work.
What are you willing to risk?
Ra's al Ghul gets the most advantage from the pit because he is perfectly willing to be insane. He is already. Any temporary disadvantage of his increased insanity is mitigated by his support staff who protect him from doing anything entire too rash during that early period. Now consider that Ra's may already be insane and thus deems any risk is better than being dead and I am certain he does his level best to avoid dying whenever possible.

Others who have been recently deceased have been brought back to life with varying results from temporary insanity to permanent insanity. Known users whose sanity is in doubt were Kobra, The Joker, King Snake, and Bane to name a few. A number of heroes have also used it including the Second Black Canary, Jason Todd and Cassandra Cain.
Limitations
A particular Lazarus Pit may only be used once, though its powers make it worth the effort. Any diseases or biological issues are erased and the body is restored to a perfect level of fitness and vigor.
The ethics on the use of the pits vary widely with villains being perfectly willing to use them while heroes questioning or even avoiding their use. For the most part, they are unknown to the general public and are rarely known even among the metahuman community as a whole.