The character of Ra's al Ghul is considered "nearly immortal" in the DC universe. He is nearly immortal because he can cheat death due to the powers of the Lazarus pit. This allows Ra's al Ghul to possess longevity, rejuvenation and youth restoration abilities. However, he can still be killed like any mortal.
Is there any evidence that Ra's al Ghul was nearly immortal in the The Dark Knight trilogy?
There is no clear evidence that he is or is not nearly immortal. We don't actually see him die at the end of Batman Begins, neither is any kind of Lazarus pit nor its absence ever mentioned in the movie.
However, there is a strong argument to be made that he is just a normal mortal person like you and me (bar any "normal" kind of better physique due to extensive training achievable by any properly trained soldier). The Nolan Batman films go to great lengths to root the whole Batman universe in a somewhat real-life scenario, where all of this could actually happen. There is stuff that's at least stretching plausibility, like advanced hightech gadgetry or the fact of someone even deciding to be such a masked vigilante. But the films themselves, their story and their whole presentation are much more rooted in the genre of the crime drama, than the one of comic superheroes. So, in the same way a Bane powered by some kind of super-soldier serum would be too much for that, a concept of a literal Lazarus pit might not fit very well there, too.
However, I deliberately said "literal", because the Nolan films, in their effort to root the ultimately comics-based Batman universe in the world of realistic crime-thrillers go to great lengths to actually transfer some of the well-known but maybe too weird concepts of the source material from their literal origins into a much more metaphorical meaning. For example, the pit where Bruce Wayne is imprisoned in the last film and that cures both his physical and psychological issues could be seen as a reference to the actual Lazarus pit, while still not being as otherworldly. Or Bane's mask that doesn't pump some weird super-soldier serum into his body, but instead makes him suspectible to much higher pain-levels than normal people by constantly delivering pain-killers to him.
And one of those instances is the immortality of Ra's al Ghul, which is repeatedly adressed throughout the film series. Just that it's not an actual literal immortality, rather than a metaphorical immortality of Ra's al Ghul and the League of Shadow's ideas. When we repeatedly see his ideology come back for Gotham through Bane or Talia, or when Bruce imagines Ra's himself in his ultimate realization in the pit. Realistically, it only is Bruce halucinating, but symbolically it is Ra's al Ghul coming back from the dead to haunt him. Ra's is very much dead, his legacy might never be. Including the concept of a literal Lazarus pit bringing his body back from the dead would undermine this whole concept.
In fact this somehow immortality while not actually being immortal is adressed in Batman Begins already, when it is revealed that Henri Ducard is Ra's al Ghul himself (rather than Ken Watanabe as we previously thought), a mere symbol he made for his ideas in the same way Bruce created a symbol to inspire Gotham city (and one that ultimately will stay immortal too):
Bruce: You're not Ra's al Ghul. I watched him die.
Ducard: But is Ra's al Ghul immortal? Are his methods supernatural?
Bruce: Or cheap parlor tricks to conceal your true identity, Ra's?
Ra's himself (or Bruce's projection of him) even alludes to his different kind of immortality in the pit scene in The Dark Knight Rises:
Ra's: Tsk, tsk, tsk. Did you not think l would return, Bruce? Hmm? I told you I was immortal.
Bruce: I watched...I watched you die.
Ra's: Oh, there are many forms of immortality.
