Does modern science, especially HeLa cells, prove that immortality (in the classic sense) is real (for these cells) or possible in the future?
In regards to HeLa cells, they are already termed immortal by biologists. According to the article:
HeLa (/ˈheɪlɑː/; also Hela or hela) is an immortal cell line used in scientific research.
So, by biological definition, they are immortal, though what that means to biologists philosophically isn't clear by the article. To restructure your assertion and presume that you mean immortality in the sense of biological organisms living without regards to temporal limit, it seems you are asking the question of whether or not a line of cells that can divide with to temporal limit implies that an organism can now live potentially without end. The simple answer is no, and here is why. It is tempting to reason as such:
P1 Cells can live potentially forever
P2 Organisms are made of cells
C Therefore, organisms can live potentially forever.
As stated, it seems persuasive, but unfortunately it is misleadingly so. This argument is an example of a well-known and classic fallacy called the fallacy of composition. Simply put, parts and wholes don't have the same properties necessarily. From the article:
The fallacy of composition arises when one infers that something is true of the whole from the fact that it is true of some part of the whole (or even of every proper part). For example: "This tire is made of rubber, therefore the vehicle to which it is a part is also made of rubber." This is fallacious, because vehicles are made with a variety of parts, many of which may not be made of rubber.
In fact, it is almost certain that the argument above is based on the fact that the verb 'to live' means two very different things for cells and organisms. A complex organism, for instance, a human being is generally recognized as having the property of a mind, and absent this property of mind, one often talks about a person being dead, but their body being alive. This of course wouldn't apply to cells from which you are reasoning. When one uses a word in two different senses, but reasons as if they are the same, one is engaged in the fallacy of equivocation:
In logic, equivocation ('calling two different things by the same name') is an informal fallacy resulting from the use of a particular word/expression in multiple senses throughout an argument leading to a false conclusion.... It is a type of ambiguity that stems from a phrase having two distinct meanings, not from the grammar or structure of the sentence.
It's easy to see why the original argument is misleading! It presumes that the lives of cells and organisms (whatever they mean exactly) essentially are the same thing. You'll be hard-pressed to find an expert in biology who will agree with that literally. As for the possiblity of organismic immortality, I doubt that any biology professional would be willing to argue that the possibility is present because of this one phenomenon. Something as complicated as living forever would require a very strong inductive argument to attain cogency.
So, to reiterate, immortal cells do not imply immortal organisms.
HeLa cells are "immortal" because their genetic machinery is damaged, causing them to reproduce all the time, and they cannot repair that damage. They are cancer cells and although you can consider them to be "alive" in the sense that they grow and divide, they are not alive in the same sense that noncancerous cells or humans for that matter are.
Haldane pointed out that when a corpse is buried, most of its cells are still alive. The corpse is not.
Living as a multicellular being involves a delicate between different cell lines, regulated by mechanisms including programmed cell death for defective cells. When these mechanisms fail, this is cancer. Individual cells become "immortal", but the organism become dead.
So if you prize immortality, envy the bacteria.